Random Variable & Discrete Distribution Random Variable - 1 1 Theoretical Probability Distribution • Discrete Probability Distributions • Continuous Probability Distributions 2 Discrete Distributions • Bernoulli • Binomial • Poisson • Geometric •… 3 Bernoulli Distribution Model (Bernoulli Probability Distribution) 4 Bernoulli Trial Definition: Bernoulli trial is a random experiment whose outcomes are classified as one of the two categories. (S , F) or (Success, Failure) or (1, 0) Example: Tossing a coin, observing Head or Tail Observing patient’s status Died or Survived. 5 Example: (Tossing a balanced coin) P(S) = P(X=1) = p = .5 P(F) = P(X=0) = 1 p = .5 Bernoulli Distribution .5 0 1 Bernoulli Probability Distribution 6 Bernoulli Probability Distribution Example: In a random experiment of casting a balanced die, we are only interested in observing 6 turns up or not. It is a Bernoulli trail. P(6) = P(X=1) = p = 1/6 P(6’) = P(X=0) =1 1/6 = 5/6 Bernoulli Distribution 0 1
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Random Variable & Discrete Distribution
Random Variable - 1
1
Theoretical Probability Distribution
• Discrete Probability Distributions
• Continuous Probability Distributions
2
Discrete Distributions
• Bernoulli • Binomial • Poisson • Geometric • …
3
Bernoulli Distribution Model (Bernoulli Probability Distribution)
4
Bernoulli Trial
Definition: Bernoulli trial is a random experiment whose outcomes are classified as one of the two categories. (S , F) or (Success, Failure) or (1, 0)
Example:
Tossing a coin, observing Head or Tail
Observing patient’s status Died or Survived.
5
Example: (Tossing a balanced coin)
P(S) = P(X=1) = p = .5
P(F) = P(X=0) = 1 p = .5
Bernoulli Distribution
.5
0 1
Bernoulli Probability Distribution
6
Bernoulli Probability Distribution
Example: In a random experiment of casting a balanced die, we are only interested in observing 6 turns up or not. It is a Bernoulli trail.
P(6) = P(X=1) = p = 1/6
P(6’) = P(X=0) =1 1/6 = 5/6
Bernoulli Distribution
0 1
Random Variable & Discrete Distribution
Random Variable - 2
7
Binomial Distribution Model (Binomial Probability Distribution)
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Binomial Experiment
A random experiment involving a sequence of independent and identical Bernoulli trials.
Example:
Toss a coin ten times, and observing Head turns up.
Roll a die 3 times, and observing a 6 turns up or not.
In a random sample of 5 from a large population, and observing subjects’ disease status. (Almost binomial)
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Binomial Probability Model
A model to find the probability of having x number successes in a sequence of n independent and identical Bernoulli trials.
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Binomial Probability Model
In a binomial experiment involving n independent and identical Bernoulli trials each with probability of success p, the probability of having x successes can be calculated with the binomial probability mass function, and it
is, for x = 0, 1, …, n,
xnx
xnx
ppx
n
ppxnx
nxXP
)1(
)1()!(!
!)(
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Factorial
n! = 1·2·3·... ·n
0! = 1
Example: 3! = 1·2·3 = 6
Example:
1012123
12345
!2)!25(
!5
2
5
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Binomial Probability
Example: A balanced die is rolled three times (or three balanced dice are rolled), what is the probability to see two 6’s?
Identify n = 3, p = 1/6, x = 2
xnx ppxnx
nxXP
)1(
)!(!
!)(
(6, 6’, 6’)
(6’, 6’, 6)
(6’, 6, 6’)
069.
6/56/13
)6/5()6/1()!23(!2
!3)2(
12
232
XP
Random Variable & Discrete Distribution
Random Variable - 3
13
Binomial Probability
Example: If 10% of the population in a community have a certain disease, what is the probability that 4 people in a random sample of 5 people from this community has the disease? (Assume binomial experiment.) Identify n = 5, p = .10, x = 4
xnx ppxnx
nxf
)1(
)!(!
!)(
0004.
9.1.5
)10.1()10(.)!45(!4
!5)(
14
454
xf
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Binomial Probability
Example: In the previous problem, what is the probability that 4 or more people have the disease?
Identify n = 5, p = .10, x = 4
0004.
00046.00001.00045.
)9(.)1(.!0!5
!5)9(.)1(.
!1!4
!5
)5()4()5()4()4(
0514
ffXPXPXP
(What is this number telling us?)
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Parameters of Binomial Distribution
Parameters of the distribution:
Mean of the distribution, = n·p
Variance of the distribution, 2 = n·p·(1 p)
Standard deviation, , is the square root of
variance.
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0 1 2 3 4 5
P(0) = .5905
P(1) = .3281
P(2) = .0729
P(3) = .0081
P(4) = .0004
P(5) = .00001
Binomial Distribution
0.590
n = 5, p = .10
= 5 x .10 = .5
2 = 5 x .1 x (1.1) = .45
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Continuous Distribution
• Normal Distribution
• Exponential Distribution
• …
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Relative Frequency Histogram
Percent
10 20 30 40 50 60 70 80 90 100 110
Test scores
Random Variable & Discrete Distribution
Random Variable - 4
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Density Curve
Density
function, f (x)
A smooth curve that fit
the distribution
Percent
Use a mathematical model to describe the variable.
10 20 30 40 50 60 70 80 90 100 110
Test scores
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Continuous Random Variable
Probability Is Area
Under Curve!
f(x)
X c d
P c X d f x dx c
d ( ) ( )
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Uniform Skewed to right Symmetrical Skewed to left
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Normal Distribution 1. ‘Bell-Shaped’ &
Symmetrical
X
f(X)
Mean
Median
Mode
2. Mean, Median, Mode Are Equal
3. Random Variable Has Infinite Range
< x <
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Normal Probability Density Function
2
2
1
e2
1)(
x
xf
f (x ) = Density of Random Variable x = Mean of the Distribution = Standard Deviation of the Distribution = 3.14159…; e = 2.71828… x = Value of Random Variable ( < x < )
Notation: N , A normal distribution with
mean and standard deviation
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Effect of Varying Parameters ( & )
X
f(X)
A C
B
Random Variable & Discrete Distribution
Random Variable - 5
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Normal Distribution Probability
dxxfdxcPd
c )()(
c dx
f(x)
Probability is
area under
curve!
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Standard Normal Distribution
Standard Normal Distribution:
A normal distribution with mean = 0 and standard deviation = 1.
Notation:
Z ~ N ( = 0, = 1)
0 Z
= 1
Cap letter Z
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Area under Standard Normal Curve
How to find the proportion of the are under the standard normal curve below z or say P ( Z < z ) = ?
Use Standard Normal Table!!!
0 z
Z
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P (1 < Z < 3)
P (Z > 3)
1 0 3 Z
0 3 Z
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Standard Normal Distribution
P(Z > 0.32) = Area above .32 = .374
0 .32
Z .00 .01 .02
0.0 .500 .496 .492
0.1 .460 .456 .452
0.2 .421 .417 .413
0.3 .382 .378 .374
Areas in the upper tail of the standard normal distribution
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Random Variable & Discrete Distribution
Random Variable - 6
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Standard Normal Distribution
P(0 < Z < 0.32) = Area between 0 and .32 = .126
0
Area = .5 - .374 = .126
.32
Z .00 .01 .02
0.0 .500 .496 .492
0.1 .460 .456 .452
0.2 .421 .417 .413
0.3 .382 .378 .374
Areas in the upper tail of the standard normal distribution
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Standard Normal Distribution
P(Z< 0.32) = Area below .32 = .626
0
Area = 1 - .374 = .626
.32
Z .00 .01 .02
0.0 .500 .496 .492
0.1 .460 .456 .452
0.2 .421 .417 .413
0.3 .382 .378 .374
Areas in the upper tail of the standard normal distribution