Math 3680 Lecture #5 Important Discrete Distributions
Example: A student randomly guesses at three questions. Each question has five possible answers, only once of which is correct. Find the probability that she gets 0, 1, 2 or 3 correct. This is the same problem as the previous one; we will now solve it by means of the binomial formula.
Example: Recall that if X ~ Binomial(3, 0.2),
P(X = 0) = 0.512
P(X = 1) = 0.384
P(X = 2) = 0.096
P(X = 3) = 0.008
Compute E(X) and SD(X).
MOMENTS OF Binomial(n, ) DISTRIBUTION
E(X) = n
SD(X) =
Var(X) =
Try this for the Binomial(3, 0.2) distribution.
Do these formulas make intuitive sense?
)1( n
)1( n
Example: A die is rolled 30 times. Let X denote the number of aces that appear.
A) Find P(X = 3).
B) Find E(X) and SD(X).
Example: Three draws are made with replacement from a box containing 6 tickets:• two labeled “1”,
• one each labeled, “2”, “3”, “4” and “5”.
Find the probability of getting two “1”s.
Example: Three draws are made without replacement from a box containing 6 tickets:• two labeled “1”,
• one each labeled, “2”, “3”, “4” and “5”.
Find the probability of getting two “1”s.
P(S1S2S3) = P(S1) P(S2 | S1) P(S3 | S1 ∩ S2) =
P(S1S2F3) = P(S1) P(S2 | S1) P(F3 | S1 ∩ S2) =
P(S1F2S3) = P(S1) P(F2 | S1) P(S3 | S1 ∩ F2) =
P(S1F2F3) = P(S1) P(F2 | S1) P(F3 | S1 ∩ F2) =
P(F1S2S3) = P(F1) P(S2 | F1) P(S3 | F1 ∩ S2) =
P(F1S2F3) = P(F1) P(S2 | F1) P(F3 | S1 ∩ F2) =
P(F1F2S3) = P(F1) P(F2 | F1) P(S3 | F1 ∩ F2) =
P(F1F2F3) = P(F1) P(F2 | F1) P(F3 | F1 ∩ F2) =
04
0
5
1
6
2
15
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4
4
5
1
6
2
15
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4
1
5
4
6
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15
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4
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5
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6
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15
1
4
1
5
2
6
4
15
3
4
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5
2
6
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15
3
4
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6
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15
3
4
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3
6
4
The Hypergeometric DistributionSuppose that n draws are made without replacement from a finite population of size N which contains G “good” objects and B = N - G “bad” objects. Let X denote the number of good objects drawn. Then
where b = n - g.
,)(
n
N
b
B
g
G
gXP
Example: Three draws are made without replacement from a box containing 6 tickets; two of which are labeled “1”, and one each labeled, “2”, “3”, “4” and “5”. Find the probability of getting two “1”’s.
MOMENTS OF
HYPERGEOMETRIC(N, G, n)
DISTRIBUTION
E(X) = n where = G / N)
SD(X) =
Var(X) =
)1(1
nN
nN
)1(1
nN
nN
REDUCTION FACTOR
The term is called the Small Population
Reduction Factor.
It always appears when we draw without replacement.
If the population is large (N > 20 n) , then the reduction factor can generally be ignored (why?).
1
N
nN
Example:
Thirteen cards are dealt from a well-shuffled deck. Let X denote the number of hearts that appear.
A) Find P(X = 3).
B) Find E(X) and SD(X).
Example. A lonely bachelor decides to play the field, deciding that a lifetime of watching “Leave It To Beaver” reruns doesn’t sound all that pleasant. On 250 consecutive days, he calls a different woman for a date. Unfortunately, through the school of hard knocks, he knows that the probability that a given woman will accept his gracious invitation is only 1%.
What is the chance that he will land
three dates?