RV and their distributions (week 5)

Question 1

Continuous Random Variable corner is

Answer 1

smooth or differentiable

Question 2

What is the derivative of CDF?

Answer 2

PDF

Question 3

Measure probability of an interval

Answer 3

use integral max b and low a.

Question 4

Properties of PDF (2)

Answer 4

1. non negative
2. area under PDF must be 1

Question 5

Expected value of continuous RV

Answer 5

E(X) = integral max infinity and min neg infinity of xf(X)dx

Question 6

Using LOTUS find g(X)

Answer 6

E(g(X)) = integral max infinity and min neg infinity of g(x)f(X)dx

Question 7

CDF of unif var

Answer 7

integral max x and min a of 1/b-a dt = (x-a)/(b-a)

min and maxnya bebas ya pokoknya dikurang di atasnya

Question 8

mean of unif var

Answer 8

(b+a)/2

Question 9

var of unif var

Answer 9

(b-a)^2/12

Question 10

location-scale transformation

Answer 10

X ~ Unif(a,b) -> Y = c+dX ~ Unif(c + ad, c +bd)

Question 11

The importance of the Uniform Dist

Answer 11

1. If U ~ Unif(0,1) then X = F^(-1) (U) is a RV with CDF F
2. If X is a RV with CDF F, then F(x) ~ Unif(0,1)

Question 12

Central Limit Theorem

Answer 12

Sum of very large number i,i,d RV is approx normal

Question 13

What is Z? given N(µ, σ)

Answer 13

(X-µ)/σ

Question 14

exponential RV?

Answer 14

E(X) = 1/lamda
Var(X) = 1/lamda^2

Question 15

What if ask
a. P(Z<1.33)
b. P(Z<-0.79)
c. P(Y <0.33) where Y = 0.5Z-1

Answer 15

a. 0.5+P(0<Z<1.33)
b. 0.5-P(0<Z<0.79)
c. Y + 1 = 0.5Z
Z = (Y+1)/0.5 = 2(y+1)
so, P(2(Y+1) < 2x(1+0.33))