Chapter 1 Random Variables

Question 1

Random variable

Answer 1

is a mapping function, mapping from a sample space to the real number line

Question 2

cdf

Answer 2

F_X(x) = P_X(X le x) for all x

Question 3

cdf is

Answer 3

right continuous

Question 4

A function F(x) is a cdf iif:

Answer 4

1. lim_x→-inf F(x) = 0
2. lim_x→inf​ F(x) = 1
3. F(x) is right continuous, ie

lim_{x→xo from right}​ F(x) = F(x_o)

Question 5

Random variables X and Y are identically distributed if

Answer 5

for every set A imo B(orell), P(X imo A) = P(Y imo A)

Question 6

F_X(x) = F_Y(x) is equivalent to

Answer 6

the random variables X and Y are identically distributed

Question 7

probability mass/density function

Answer 7

pmf: discrete; f_X(x) = P(X=x)
pdf: continuous; F_X(x) = integrand f_X(t) dt evaluated from -inf to x

Question 8

to be a valid pdf or pmf

Answer 8

a. f_X(x) ge 0 for all x
b. sum f_X(x) = 1 or integral of f_X(x) dx =1 over all x