Module 9: Important Discrete Random Variables

Question 1

whats a bernoulli random variable?

Answer 1

a single experiment is performed. It results in success with probability p, 0 < p < 1, and failure otherwise. X defined as the number of successes is called the Bernoulli random variable (with parameter p) and satisfies:

p(x) = p^x * (1-p)^(1-x), x = 0, 1

E(X) = p and Var(X) = p(1-p)

Question 2

what is the linearity of expectation and variance for independent random variables?

Answer 2

E(Y_1 + Y_2) = E(Y_1) + E(Y_2)
Var(Y_1 + Y_2) = Var(Y_1) + Var(Y_2)

Question 3

whats the taylor series def for e^x?

Answer 3

e^x = SUM((x^n)/n!) from n=0 to infinity

Question 4

what is the binomial random variable?

Answer 4

there are n independent Bernoulli experiments are performed, each resulting in success with a fixed probability p, 0 < p < 1. X defined as a the number of successes in in the n trails is called the binomial random variable (with parameters n,p) and satisfies:

p(x) = c(n,x) * p^x * (1-p)^n-x, x = 0, 1, 2, … , n

E(X) = np
Var(X) = np(1 - p)

Question 5

what is the Poisson Random Variable?

Answer 5

n independent Bernoulli experiments are performed, each resulting in success with a fixed probability p, 0 < p < 1. X defined as the number of successes in the n trails is binomial and can be approximated by the Poisson random variable with parameter λ = np > 0 when n is large enough and p is small enough to make λ moderate. A possion random variable has the following features:

p(x) = (λ^x)/x! * e^-λ, x = 0,1,2,…

E(X) = λ and Var(X) = λ

Note that λ = np should be interpreted as the average number of successes per unit time of space.