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18.440: Lecture 19 Normal random variables Scott Sheffield MIT 18.440 Lecture 19
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18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

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Page 1: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

18.440: Lecture 19

Normal random variables

Scott Sheffield

MIT

18.440 Lecture 19

Page 2: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 3: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 4: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Suppose we toss a million fair coins. How many heads will weget?

I About half a million, yes, but how close to that? Will we beoff by 10 or 1000 or 100,000?

I How can we describe the error?

I Let’s try this out.

18.440 Lecture 19

Page 5: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Suppose we toss a million fair coins. How many heads will weget?

I About half a million, yes, but how close to that? Will we beoff by 10 or 1000 or 100,000?

I How can we describe the error?

I Let’s try this out.

18.440 Lecture 19

Page 6: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Suppose we toss a million fair coins. How many heads will weget?

I About half a million, yes, but how close to that? Will we beoff by 10 or 1000 or 100,000?

I How can we describe the error?

I Let’s try this out.

18.440 Lecture 19

Page 7: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Suppose we toss a million fair coins. How many heads will weget?

I About half a million, yes, but how close to that? Will we beoff by 10 or 1000 or 100,000?

I How can we describe the error?

I Let’s try this out.

18.440 Lecture 19

Page 8: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 9: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 10: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 11: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 12: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 13: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 14: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 15: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Tossing coins

I Toss n coins. What is probability to see k heads?

I Answer: 2−k(nk

).

I Let’s plot this for a few values of n.

I Seems to look like it’s converging to a curve.

I If we replace fair coin with p coin, what’s probability to see kheads.

I Answer: pk(1− p)n−k(nk

).

I Let’s plot this for p = 2/3 and some values of n.

I What does limit shape seem to be?

18.440 Lecture 19

Page 16: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 17: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 18: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.I Happens to be a nice trick. Write I =

∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.

18.440 Lecture 19

Page 19: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.I Happens to be a nice trick. Write I =

∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.

18.440 Lecture 19

Page 20: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.

I Happens to be a nice trick. Write I =∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.

18.440 Lecture 19

Page 21: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.I Happens to be a nice trick. Write I =

∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.

I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.

18.440 Lecture 19

Page 22: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.I Happens to be a nice trick. Write I =

∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.

18.440 Lecture 19

Page 23: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable

I Say X is a (standard) normal random variable iffX (x) = f (x) = 1√

2πe−x

2/2.

I Clearly f is always non-negative for real values of x , but howdo we show that

∫∞−∞ f (x)dx = 1?

I Looks kind of tricky.I Happens to be a nice trick. Write I =

∫∞−∞ e−x

2/2dx . Then

try to compute I 2 as a two dimensional integral.I That is, write

I 2 =

∫ ∞−∞

e−x2/2dx

∫ ∞−∞

e−y2/2dy =

∫ ∞−∞

∫ ∞−∞

e−x2/2dxe−y

2/2dy .

I Then switch to polar coordinates.

I 2 =

∫ ∞0

∫ 2π

0e−r

2/2rdθdr = 2π

∫ ∞0

re−r2/2dr = −2πe−r

2/2∣∣∣∞0,

so I =√

2π.18.440 Lecture 19

Page 24: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 25: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 26: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 27: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 28: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 29: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Standard normal random variable mean and variance

I Say X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I Question: what are mean and variance of X?

I E [X ] =∫∞−∞ xf (x)dx . Can see by symmetry that this zero.

I Or can compute directly:

E [X ] =

∫ ∞−∞

1√2π

e−x2/2xdx =

1√2π

e−x2/2∣∣∣∞−∞

= 0.

I How would we computeVar[X ] =

∫f (x)x2dx =

∫∞−∞

1√2πe−x

2/2x2dx?

I Try integration by parts with u = x and dv = xe−x2/2dx .

Find that Var[X ] = 1√2π

(−xe−x2/2∣∣∣∞−∞

+∫∞−∞ e−x

2/2dx) = 1.

18.440 Lecture 19

Page 30: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

General normal random variables

I Again, X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I What about Y = σX + µ? Can we “stretch out” and“translate” the normal distribution (as we did last lecture forthe uniform distribution)?

I Say Y is normal with parameters µ and σ2 iff (x) = 1√

2πσe−(x−µ)

2/2σ2.

I What are the mean and variance of Y ?

I E [Y ] = E [X ] + µ = µ and Var[Y ] = σ2Var[X ] = σ2.

18.440 Lecture 19

Page 31: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

General normal random variables

I Again, X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I What about Y = σX + µ? Can we “stretch out” and“translate” the normal distribution (as we did last lecture forthe uniform distribution)?

I Say Y is normal with parameters µ and σ2 iff (x) = 1√

2πσe−(x−µ)

2/2σ2.

I What are the mean and variance of Y ?

I E [Y ] = E [X ] + µ = µ and Var[Y ] = σ2Var[X ] = σ2.

18.440 Lecture 19

Page 32: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

General normal random variables

I Again, X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I What about Y = σX + µ? Can we “stretch out” and“translate” the normal distribution (as we did last lecture forthe uniform distribution)?

I Say Y is normal with parameters µ and σ2 iff (x) = 1√

2πσe−(x−µ)

2/2σ2.

I What are the mean and variance of Y ?

I E [Y ] = E [X ] + µ = µ and Var[Y ] = σ2Var[X ] = σ2.

18.440 Lecture 19

Page 33: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

General normal random variables

I Again, X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I What about Y = σX + µ? Can we “stretch out” and“translate” the normal distribution (as we did last lecture forthe uniform distribution)?

I Say Y is normal with parameters µ and σ2 iff (x) = 1√

2πσe−(x−µ)

2/2σ2.

I What are the mean and variance of Y ?

I E [Y ] = E [X ] + µ = µ and Var[Y ] = σ2Var[X ] = σ2.

18.440 Lecture 19

Page 34: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

General normal random variables

I Again, X is a (standard) normal random variable iff (x) = 1√

2πe−x

2/2.

I What about Y = σX + µ? Can we “stretch out” and“translate” the normal distribution (as we did last lecture forthe uniform distribution)?

I Say Y is normal with parameters µ and σ2 iff (x) = 1√

2πσe−(x−µ)

2/2σ2.

I What are the mean and variance of Y ?

I E [Y ] = E [X ] + µ = µ and Var[Y ] = σ2Var[X ] = σ2.

18.440 Lecture 19

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Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

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Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 37: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 38: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 39: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 40: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 41: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Cumulative distribution function

I Again, X is a standard normal random variable iff (x) = 1√

2πe−x

2/2.

I What is the cumulative distribution function?

I Write this as FX (a) = P{X ≤ a} = 1√2π

∫ a−∞ e−x

2/2dx .

I How can we compute this integral explicitly?

I Can’t. Let’s just give it a name. WriteΦ(a) = 1√

∫ a−∞ e−x

2/2dx .

I Values: Φ(−3) ≈ .0013, Φ(−2) ≈ .023 and Φ(−1) ≈ .159.

I Rough rule of thumb: “two thirds of time within one SD ofmean, 95 percent of time within 2 SDs of mean.”

18.440 Lecture 19

Page 42: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 43: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Outline

Tossing coins

Normal random variables

Special case of central limit theorem

18.440 Lecture 19

Page 44: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 45: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 46: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 47: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 48: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 49: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

Page 50: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

DeMoivre-Laplace Limit Theorem

I Let Sn be number of heads in n tosses of a p coin.

I What’s the standard deviation of Sn?

I Answer:√npq (where q = 1− p).

I The special quantity Sn−np√npq describes the number of standard

deviations that Sn is above or below its mean.

I What’s the mean and variance of this special quantity? Is itroughly normal?

I DeMoivre-Laplace limit theorem (special case of centrallimit theorem):

limn→∞

P{a ≤ Sn − np√npq

≤ b} → Φ(b)− Φ(a).

I This is Φ(b)− Φ(a) = P{a ≤ X ≤ b} when X is a standardnormal random variable.

18.440 Lecture 19

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Problems

I Toss a million fair coins. Approximate the probability that Iget more than 501, 000 heads.

I Answer: well,√npq =

√106 × .5× .5 = 500. So we’re asking

for probability to be over two SDs above mean. This isapproximately 1− Φ(2) = Φ(−2) ≈ .159.

I Roll 60000 dice. Expect to see 10000 sixes. What’s theprobability to see more than 9800?

I Here√npq =

√60000× 1

6 ×56 ≈ 91.28.

I And 200/91.28 ≈ 2.19. Answer is about 1− Φ(−2.19).

18.440 Lecture 19

Page 52: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Problems

I Toss a million fair coins. Approximate the probability that Iget more than 501, 000 heads.

I Answer: well,√npq =

√106 × .5× .5 = 500. So we’re asking

for probability to be over two SDs above mean. This isapproximately 1− Φ(2) = Φ(−2) ≈ .159.

I Roll 60000 dice. Expect to see 10000 sixes. What’s theprobability to see more than 9800?

I Here√npq =

√60000× 1

6 ×56 ≈ 91.28.

I And 200/91.28 ≈ 2.19. Answer is about 1− Φ(−2.19).

18.440 Lecture 19

Page 53: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Problems

I Toss a million fair coins. Approximate the probability that Iget more than 501, 000 heads.

I Answer: well,√npq =

√106 × .5× .5 = 500. So we’re asking

for probability to be over two SDs above mean. This isapproximately 1− Φ(2) = Φ(−2) ≈ .159.

I Roll 60000 dice. Expect to see 10000 sixes. What’s theprobability to see more than 9800?

I Here√npq =

√60000× 1

6 ×56 ≈ 91.28.

I And 200/91.28 ≈ 2.19. Answer is about 1− Φ(−2.19).

18.440 Lecture 19

Page 54: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Problems

I Toss a million fair coins. Approximate the probability that Iget more than 501, 000 heads.

I Answer: well,√npq =

√106 × .5× .5 = 500. So we’re asking

for probability to be over two SDs above mean. This isapproximately 1− Φ(2) = Φ(−2) ≈ .159.

I Roll 60000 dice. Expect to see 10000 sixes. What’s theprobability to see more than 9800?

I Here√npq =

√60000× 1

6 ×56 ≈ 91.28.

I And 200/91.28 ≈ 2.19. Answer is about 1− Φ(−2.19).

18.440 Lecture 19

Page 55: 18.440: Lecture 19 .1in Normal random variablesmath.mit.edu/~sheffield/440/Lecture19.pdf · 2014-02-11 · Outline Tossing coins Normal random variables Special case of central limit

Problems

I Toss a million fair coins. Approximate the probability that Iget more than 501, 000 heads.

I Answer: well,√npq =

√106 × .5× .5 = 500. So we’re asking

for probability to be over two SDs above mean. This isapproximately 1− Φ(2) = Φ(−2) ≈ .159.

I Roll 60000 dice. Expect to see 10000 sixes. What’s theprobability to see more than 9800?

I Here√npq =

√60000× 1

6 ×56 ≈ 91.28.

I And 200/91.28 ≈ 2.19. Answer is about 1− Φ(−2.19).

18.440 Lecture 19