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Bernoulli and Binomial Distributions
Bernoulli Distribution Binomial Distribution Exercises
Bernoulli Distribution
a flipped coin turns up either heads or tailsan item on an assembly line is either defective or not defectivea piece of fruit is either damaged or not damageda cow is either pregnant or not pregnanta child is either female or male
X =
{1, if the outcome of the trial is a success0, if the outcome of the trial is a failure
X =
{1, w.p. p0, w.p. q = 1− p
We can write the pmf as
p(x) = px(1− p)1−x, x = 0, 1
Arthur Berg Bernoulli and Binomial Distributions 2/ 9
Bernoulli Distribution Binomial Distribution Exercises
Bernoulli Distribution
a flipped coin turns up either heads or tailsan item on an assembly line is either defective or not defectivea piece of fruit is either damaged or not damageda cow is either pregnant or not pregnanta child is either female or male
X =
{1, if the outcome of the trial is a success0, if the outcome of the trial is a failure
X =
{1, w.p. p0, w.p. q = 1− p
We can write the pmf as
p(x) = px(1− p)1−x, x = 0, 1
Arthur Berg Bernoulli and Binomial Distributions 2/ 9
Bernoulli Distribution Binomial Distribution Exercises
Bernoulli Distribution
a flipped coin turns up either heads or tailsan item on an assembly line is either defective or not defectivea piece of fruit is either damaged or not damageda cow is either pregnant or not pregnanta child is either female or male
X =
{1, if the outcome of the trial is a success0, if the outcome of the trial is a failure
X =
{1, w.p. p0, w.p. q = 1− p
We can write the pmf as
p(x) = px(1− p)1−x, x = 0, 1
Arthur Berg Bernoulli and Binomial Distributions 2/ 9
Bernoulli Distribution Binomial Distribution Exercises
Bernoulli Distribution
a flipped coin turns up either heads or tailsan item on an assembly line is either defective or not defectivea piece of fruit is either damaged or not damageda cow is either pregnant or not pregnanta child is either female or male
X =
{1, if the outcome of the trial is a success0, if the outcome of the trial is a failure
X =
{1, w.p. p0, w.p. q = 1− p
We can write the pmf as
p(x) = px(1− p)1−x, x = 0, 1
Arthur Berg Bernoulli and Binomial Distributions 2/ 9
Bernoulli Distribution Binomial Distribution Exercises
Expectation and Variance of Bernoulli(p)
E(X) =1∑
x=0
xp(x) = 0p(0) + 1p(1) = 0(1− p) + 1(p) = p
Noting that when X ∼ Bernoulli(p), X2 ∼ X, i.e. “X2 has the same distributionas X”.
var(X) = E(X2)− [E(X)]2 = p− p2 = p(1− p) = pq
Arthur Berg Bernoulli and Binomial Distributions 3/ 9
Bernoulli Distribution Binomial Distribution Exercises
Expectation and Variance of Bernoulli(p)
E(X) =1∑
x=0
xp(x) = 0p(0) + 1p(1) = 0(1− p) + 1(p) = p
Noting that when X ∼ Bernoulli(p), X2 ∼ X, i.e. “X2 has the same distributionas X”.
var(X) = E(X2)− [E(X)]2 = p− p2 = p(1− p) = pq
Arthur Berg Bernoulli and Binomial Distributions 3/ 9
Bernoulli Distribution Binomial Distribution Exercises
Expectation and Variance of Bernoulli(p)
E(X) =1∑
x=0
xp(x) = 0p(0) + 1p(1) = 0(1− p) + 1(p) = p
Noting that when X ∼ Bernoulli(p), X2 ∼ X, i.e. “X2 has the same distributionas X”.
var(X) = E(X2)− [E(X)]2 = p− p2 = p(1− p) = pq
Arthur Berg Bernoulli and Binomial Distributions 3/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Binomial Distribution
We are typically interested in n independent Bernoulli trials, each with aprobability p of success.Let Y1, Y2, . . . , Yn denote independent and identically-distributed (iid)Bernoulli(p) random variables.The sum X =
∑ni=1 Yi denotes the number of successes among n sampled items.
X is defined to be the binomial distribution with n trials and probability p ofsuccess, i.e. X ∼ binomial(n, p).The pmf of the binomial(n, p) is
p(x) =(
nx
)px(1− p)n−x =
(nx
)pxqn−x
Arthur Berg Bernoulli and Binomial Distributions 4/ 9
Bernoulli Distribution Binomial Distribution Exercises
Checking the Probability Mass Function Sums to One
The pmf of any discrete random variable should sum to one.Recall the binomial theorem:
(x + y)n =n∑
i=0
(ni
)xnyn−i
Thereforen∑
x=0
p(x) =n∑
x=0
(nx
)px(1− p)n−x = p + (1− p)n = 1n = 1
Arthur Berg Bernoulli and Binomial Distributions 5/ 9
Bernoulli Distribution Binomial Distribution Exercises
Checking the Probability Mass Function Sums to One
The pmf of any discrete random variable should sum to one.Recall the binomial theorem:
(x + y)n =n∑
i=0
(ni
)xnyn−i
Thereforen∑
x=0
p(x) =n∑
x=0
(nx
)px(1− p)n−x = p + (1− p)n = 1n = 1
Arthur Berg Bernoulli and Binomial Distributions 5/ 9
Bernoulli Distribution Binomial Distribution Exercises
Checking the Probability Mass Function Sums to One
The pmf of any discrete random variable should sum to one.Recall the binomial theorem:
(x + y)n =n∑
i=0
(ni
)xnyn−i
Thereforen∑
x=0
p(x) =n∑
x=0
(nx
)px(1− p)n−x = p + (1− p)n = 1n = 1
Arthur Berg Bernoulli and Binomial Distributions 5/ 9
Bernoulli Distribution Binomial Distribution Exercises
Checking the Probability Mass Function Sums to One
The pmf of any discrete random variable should sum to one.Recall the binomial theorem:
(x + y)n =n∑
i=0
(ni
)xnyn−i
Thereforen∑
x=0
p(x) =n∑
x=0
(nx
)px(1− p)n−x = p + (1− p)n = 1n = 1
Arthur Berg Bernoulli and Binomial Distributions 5/ 9
Bernoulli Distribution Binomial Distribution Exercises
Mean and Variance of Binomial(n, p)
Recall X ∼ binomial(n, p) can be written as X =∑n
i=1 Yi where
Y1, . . . , Yniid∼ Bernoulli(p).
Therefore
E(X) = E
(n∑
i=1
Yi
)=
n∑i=1
E(Yi) =n∑
i=1
p = np
And because of the independence, we similarly calculate the variance to be
var(X) = var
(n∑
i=1
Yi
)=
n∑i=1
var(Yi) =n∑
i=1
pq = npq
Arthur Berg Bernoulli and Binomial Distributions 6/ 9
Bernoulli Distribution Binomial Distribution Exercises
Mean and Variance of Binomial(n, p)
Recall X ∼ binomial(n, p) can be written as X =∑n
i=1 Yi where
Y1, . . . , Yniid∼ Bernoulli(p).
Therefore
E(X) = E
(n∑
i=1
Yi
)=
n∑i=1
E(Yi) =n∑
i=1
p = np
And because of the independence, we similarly calculate the variance to be
var(X) = var
(n∑
i=1
Yi
)=
n∑i=1
var(Yi) =n∑
i=1
pq = npq
Arthur Berg Bernoulli and Binomial Distributions 6/ 9
Bernoulli Distribution Binomial Distribution Exercises
Mean and Variance of Binomial(n, p)
Recall X ∼ binomial(n, p) can be written as X =∑n
i=1 Yi where
Y1, . . . , Yniid∼ Bernoulli(p).
Therefore
E(X) = E
(n∑
i=1
Yi
)=
n∑i=1
E(Yi) =n∑
i=1
p = np
And because of the independence, we similarly calculate the variance to be
var(X) = var
(n∑
i=1
Yi
)=
n∑i=1
var(Yi) =n∑
i=1
pq = npq
Arthur Berg Bernoulli and Binomial Distributions 6/ 9
Bernoulli Distribution Binomial Distribution Exercises
Problem From Last Time
discrete r.v.’s with mean 0 and variance 1Consider the discrete random variable
X =
{a, w.p. pb, w.p. 1− p
In class, we saw that a = ±1, b = ∓1, and p = 1/2 we have E(X) = 0 andvar(X) = 1.Are there other choices of a, b, and p that will give E(X) = 0 and var(X) = 1?
Arthur Berg Bernoulli and Binomial Distributions 7/ 9
Bernoulli Distribution Binomial Distribution Exercises
Exercise 4.45 (p.147)
Arthur Berg Bernoulli and Binomial Distributions 8/ 9
Bernoulli Distribution Binomial Distribution Exercises
Exercise 4.55 (p.149)
Arthur Berg Bernoulli and Binomial Distributions 9/ 9