Consider a random variable X whose distribution function cdf

Consider a random variable X whose distribution function (cdf) is given by:

FX(x) =

0 if x < 2

0.1 if 2 x < 1.1

0.3 if 1.1 x < 2

0.6 if 2 x < 3

1 if x 3

(a) Give the probability mass function, p(x), of X, explicitly.

(b) Compute P(2 < X < 3).

(c) Compute P(X 3).

(d) Compute P(X 3 | X 0).

(e) What is the cdf (distribution function) of Y = X2 ? (be explicit!)

(f) Compute E(X). Also compute E(X3 cos X).

Solution

a)

STEP 1 FIND THE PDF

for -2<x<1.1 is the same 0.1

for 1.1<x<2 will be 0.3 - 0.1 = 0.2

for 2<x<3 will be 0.6-0.3 = 0.3

for x>3 will be 1 - 0.6 = 0.4

pmf will be that values multiply for X in each case X will be different

b)

P(2<x<3) = 0.3

c)

P(x>3) = 0.4

d)

P( x>3 / x>0) = 0.2 / 0.4 = 0.5

for the others literal I can gladly help you but

you should post it in a new question