Chapter 4

Question 1

If X is a discrete random variable having a probability mass function p(x), then the expectation, or the expected value, of X, denoted by E[X], is defined by

Answer 1

The sum such that x:p(x)>0 of xp(x)

Question 2

If X is a random variable with mean E[X], then the variance of X, denoted by Var(X), is defined by

Answer 2

Var(X) = E[X²] − (E[X])²

Question 3

E[X] of a Poisson random variable

Answer 3

E[X] = λ

Question 4

Var(X) of a Poisson random variable

Answer 4

Var(X) = λ

Question 5

Parameter for a Poisson random variable

Answer 5

λ

Question 6

Range of a Poisson random variable

Answer 6

λ > 0

Question 7

pmf of a Poisson random variable

Answer 7

p(k) = P{X = k} = (λ^k/k!)e^-λ​

Question 8

E[X] of a Binomial random variable

Answer 8

E[X] = np

Question 9

Var(X) of a Binomial random variable

Answer 9

Var(X) = np(1 - p)

Question 10

Parameters for a Binomial random variable

Answer 10

n — number of trials

p — success probability in each trial

Question 11

Range of a Binomial random variable

Answer 11

n ∈ N₀

p ∈ [0,1]

Question 12

pmf of a Binomial random variable

Answer 12

(n k) p^k (1 - p)^{n - k}

Question 13

E[X] of a Geometric random variable

Answer 13

E[X] = 1/p

Question 14

Var(X) of a Geometric random variable

Answer 14

Var(X) = (1 - p)/p²

Question 15

Parameters of a Geometric random variable

Answer 15

p - probability

k - trials

Question 16

Range of a Geometric random variable

Answer 16

0 < p <= 1

k ∈ {1, 2, 3,…}

Question 17

pmf of a Geometric random variable

Answer 17

(1 - p)^{k - 1}p

Question 18

E[X] of a Negative Binomial

Answer 18

E[X] = (pr)/(1 - p)

Question 19

Var(X) of a Negative Binomial

Answer 19

(pr)/(1 - p)²

Question 20

Parameters for Negative Binomial

Answer 20

r — number of failures until the experiment is stopped

p — success probability in each experiment

k — number of successes

Question 21

Range of Negative Binomial

Answer 21

r > 0

p ∈ (0,1)

k ∈ { 0, 1, 2, 3, … }

Question 22

pmf of Negative Binomial

Answer 22

((k + r - 1) choose k) · (1 - p)^rp^k