3. Discrete Random Variables

Question 1

What is a pmf?

Answer 1

For a discrete random variable X, we define the probability mass function (p.m.f. for short) pX to be pX(x) := P(X = x).

1. pX(x) ≥ 0 ∀x ∈ R.
2. Probability mass functions must ‘sum to 1’

Question 2

What is a cdf?

Answer 2

We define the cumulative distribution function, FX of a random variable X (discrete or continuous) to be FX(x) := P(X ≤ x).

Question 3

what is the cdf in terms of the pdf?

Answer 3

FX(x) := P(X ≤ x) = the sum from a≤x,a∈RX of all pX(a).

Question 4

What is a quantile and percentile?

Answer 4

For α ∈ [0, 1] the α quantile (or 100 × α percentile) is the
smallest value of x such that FX(x) ≥ α.
The median is the 0.5 quantile.

Question 5

Define expectation (mean).

Answer 5

The expectation E(X) of a discrete random variable X is
defined as
E(X) := the sum from (x∈RX) of all xP(X = x).

Question 6

Define Variance

Answer 6

Var(X) : = E{X − E(X)}^2 = E{(X − µX)^2}.

Var(X) = E(X^2) − E(X)^2

Question 7

E(aX+b) and Var(aX+b)

Answer 7

E(aX+b) =aE(x) + b

Var(aX+b) = a^2Var(X)

Question 8

Bernoulli Distribution

Answer 8

A Bernoulli r.v. can take values of 0 and 1 only. px(1) = p.

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

Question 9

Binomial Distribution

Answer 9

pmf: pX(x) = (n! / x!(n − x)!) p^x (1 − p)^(n−x).
E(X) = np
Var(X) = np(1 − p)

Question 10

Poisson Distribution

Answer 10

Used to represent count data: to number of times something happens within some finite interval.
pX(x) = e^(−λ) λ^x / x!
E(X) = Var(X) = λ.

Question 11

Geometric Distribution

Answer 11

pX(x) = (1 − p)^(x−1) p
FX(x) = 1 − (1 − p)^x
E(X) = 1 / p
Var(X) = (1 − p) / p^2.

Question 12

Joint pmf

Answer 12

P{(X=x)∩(Y=y)}

Question 13

Joint cdf

Answer 13

P{(X ≤ x) ∩ (Y ≤ y)}

Question 14

The expectation of the sum of two random variables E(X+Y) is

Answer 14

E(X) + E(Y)

Question 15

Covariance

Answer 15

The covariance between two random variables X and Y is
Cov(X, Y) = E{(X-µx)(Y-µy)}
or, Cov(X, Y) = E(XY) - E(X)E(Y)

Question 16

Correlation

Answer 16

Cor(X, Y ) := Cov(X, Y) / sqrt(Var(X) Var(Y))

Question 17

What is the variance of X + Y, where X and Y are random variables?

Answer 17

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)