Chapter 3

Question 1

random variable

Answer 1

a random variable is a real-valued function that assigns a numerical valueto each possible outcome of the random experiment

The range of a random variable X
, shown by Range (X) or RX, is the set of possible values of X

Question 2

discrete random variable

Answer 2

X is a discrete random variable, if its range is countable.

Question 3

probability mass function (PMF)

Answer 3

Let X be a discrete random variable with range RX = {x1,x2,x3,…} (finite or
countably infinite). The function

PX(xk) = P(X=xk), for k = 1,2,3,…,

is called the probability mass function of X

Question 4

probability distribution

Answer 4

is the same as the PMF, but for discrete random variables

Question 5

Bernoulli distribution

Answer 5

A random variable X is said to be a Bernoulli random variable with paramter p shown as X ∼ Bernoulli(p), if its PMF is given by

PX(x) = (p for x = 1, 1-p for x = 0 and 0 for anything else}

where 0 < p < 1

Question 6

indicator random variable

Answer 6

the indicator random variable IA for an event A is defined by

IA = {1 if the event A occurs, 0 otherwise}

IA ∼ Bernoulli (P(A))

Question 7

geometric random variable

Answer 7

A random variable X is said to be a geometric random variable with parameter p, shown as X ∼ Geometric(p), if its PMF is given by

PX(k) = {p(1-p)^k-1 for k = 1,2,3,…
or 0 otherwise}

Question 8

cumulsyive distribution function (CDF)

Answer 8

The cumulative distribution function (CDF) of random variable X is defined as

FX(x)=P(X≤x), for all x∈R.

Question 9

Cumulative distribution function (CDF)

Answer 9

The cumulative distribution function (CDF) of random variable X is defined as:

FX(x)=P(X≤x), for all x∈R.

Question 10

expected value (=mean=average)

Answer 10

Let X be a discrete random variable with range RX={x1,x2,x3,…} (finite or countably infinite). The expected value of X, denoted by EX is defined as

EX = ∑xk∈RX xkP(X=xk) = ∑xk∈RX xkPX(xk).

Question 11

Law of the unconscious statistician (LOTUS) for discrete random variables

Answer 11

E[g(X)]=∑xk∈RX g(xk)PX(xk)

Question 12

variance

Answer 12

The variance is a measure of how spread out the distribution of a random variable is.

The variance of a random variable X, with mean EX=μX, is defined as

Var(X)=E[(X−μX)^2].

Question 13

standard deviation

Answer 13

The standard deviation of a random variable X is defined as

SD(X) = σX = √(Var(X))

Question 14

computational formula for the variance

Answer 14

Var(X) = E[X^2] - [EX]^2