Random Variables

Question 1

What is an RV X(w)

Answer 1

An RV X on R is a mapping from omega to R s.t X <= x = (w in omega : X(w) <= x) E F

Question 2

What is equivalent definition of an RV

Answer 2

Equivalent definition of an RV is:
For all A in B, X^-1[A) = (w in omega : X(w) in A) E F
X : (omega, F) to (R, B(R))
Set, sigma-field on omega, set, sigma-field

Question 3

How can we make an RV X infinite

Answer 3

We can make an RV X infinite by extending definition of RV to R U (inf) U (-inf)

Question 4

What is the law of an RV X

Answer 4

Law of an RV C is a probability measure Px in (R, B(R)) defined by Px(A) = P(X in A) for all A in B(R)

Question 5

What is cumulative distribution function

Answer 5

Cumulative distribution function FX of X us map from R to [0,1] defined by Fx = P(X <= x) = Px((-inf,]]

Question 6

What are lim x tends to -inf Fx(u) and lim x tends to inf Fx(X)

Answer 6

Lim x tends to -inf Fx(u) = 0 and lim x tends to inf Fx(X) = 1

Question 7

Where is X a discrete RV

Answer 7

X is a discrete RV when X(omega) = ( X(w) : w in omega) is countable

Question 8

What is PMF

Answer 8

PMF is function defined by pX(x) = P(X=x) for all x in R

Question 9

When is X continuous

Answer 9

X is continuous RV if exists a non-negative function fX : R to R+ s.t P(a < X <= b) = integral b Down to a fX(x) dx = Fx(b) - Fx(a) (CDF)
fx is called density of X