Random Variables

Question 1

Discrete Random variables

Answer 1

A random variable that can take on at most a countable number of possible values is said to be discrete

Question 2

Expected Value

Answer 2

E[X] = ∑xip(xi) xp(x)

Question 3

Var(X)

Answer 3

Var(X) = E[(X - µ)²]

Var(X) = E[X²] - (E[X])²

Question 4

Bernoulli Random Variable

Answer 4

Random variable given by equation:

p(0) = P{X=0} = 1 - p

p(1) = P{X=1} = p

Question 5

Poisson random variable

Answer 5

p(i) = e^-λ * (λⁱ)/i!

λ = np

Question 6

E[X]

Answer 6

= ∑ xi.p(xi)

Question 7

E[g(X)]

Answer 7

= ∑ g(xi).p(xi)

Question 8

Corollary 4.1

Answer 8

If a and b are constants, then E[aX + b] = aE[X] + b

Question 9

First moment of X

Answer 9

E[X] (the mean)

Question 10

Nth moment of X

Answer 10

E[X^n], n ≥ 1

Question 11

E[X^n]

Answer 11

∑ x^np(x)

Question 12

E[X], where X is a binomial random variable with parameters n and p

Answer 12

E[x] = np

Question 13

Var(X), where X is a binomial random variable with parameters n and p

Answer 13

Var(X) = np(1 - p)

Question 14

Proposition 6.1

Answer 14

If X is a binomial random variable with parameters (n, p) where 0 < p < q, then as k goes from 0 to n, P{ X = k } first increases monotonically and then decreases monotonically, reaching its largest value when k is the largest integer less than or equal to (n + 1)p

Question 15

Poisson distribution requirements

Answer 15

• Events are occurring independently - The probability that an event occurs in a given length of time does not change through time.

Question 16

Variance for a poisson distribution

Answer 16

equals the mean/expected value = µ = λ

Question 17

Geometric random variable

Answer 17

Suppose that independent trials, (each having a probability p, 0

<1, of being a success,) are performed until a success occurs. If we let X equal the number of trials, then: P{X = n} = p(1 - p)^(n-1), n = 1,2,…

Question 18

Negative Binomial Random Variable

Answer 18

Suppose that independent trials are performed until a total of r successes is accumulated. If we let X equal the number of trials required, then: P{X = n} = (n-1)C(r-1) p^r(1 - p)^(n-r)

Question 19

Requirements for the Negative Binomial Random Variable

Answer 19

• Independent trials - Each trial results one of 2 outcomes: success or failure - P(success) = p, this stays constant throughout - P(failure) = 1 - p - X represents the trial number of the rth success

Question 20

Mean of a Negative Binomial Random Variable

Answer 20

µ = r/p

Question 21

Variance of a negative binomial random variable

Answer 21

r(1 - p)/p^2