Families Of Continuous RVs

Question 1

What is the CDF of uniform distribution

Answer 1

CDF of uniform distribution is x-a/(b-a) if a< = x < = b
0 if x < a
1 if x > b

Question 2

If X = a +(b-a)U, then what is distribution of X

Answer 2

If X = a +(b-a)U, then distribution of X is X~(a,b)

Question 3

What are expectation and variance of X~U(a,b)

Answer 3

Expectation and variance of X~U(a,b) are:
E(X) = 0.5(a+b)
Var(X)=(b-a)^2 / 12

Question 4

What is PDF of normal distribution

Answer 4

PDF of normal distribution is:

1/root(2pivariance) *e^-(x-u)^2 /2var

Question 5

What does the de Moivre laplace theorem state

Answer 5

The de Moivre laplace theorem states that for S~B(n,p) for any a
P(Sn-np /root(np(1-p)) < = a) tends to PHI(a) as n tends to infinity. PHI(a) = CDF of standard normal

Question 6

What is PDF of exponential distribution

Answer 6

PDF of exponential distribution is:

fx = Le^-Lx x > = 0 and 0 otherwise

Question 7

What is CDF of exponential distribution

Answer 7

CDF of exponential distribution is:

Fx = 1-e^-Lx for x >= 0 and 0 otherwise

Question 8

What are Mean and variance for exponential distribution

Answer 8

Mean and variance for exponential distribution are:
E(X)= 1/L
Var(X) = 1/L^2

Question 9

What is the memory less property of the exponential distribution

Answer 9

Memory less property of the exponential distribution is:
P(X > t+s | X > t) = e^-Ls
Called memory less as it doesn’t depend on t

Question 10

What is hazard rate

Answer 10

Hazard rate is:

H(t) = fx(t) /1-Fx(t) for a RV X at a time t