Families Of Continuous RVs 2

Question 1

What is PDF of weibull distribution

Answer 1

PDF of weibull distribution is:

fx = LB * x^B-1 * e^-Lx^B for x >= 0 and 0 otherwise

Question 2

What is the CDF of weibull distribution

Answer 2

CDF of weibull distribution is:

Fx = 1-e^-L(x^B) for x >= 0 and 0 otherwise

Question 3

What is the gamma function

Answer 3

The gamma function is defined for t > 0 as:
G(t) = integral infinity down to 0 x^t-1 * e^-x dx
G(0.5) = root(pi)

Question 4

What is G(t)

Answer 4

G(t) = (t-1)! For integer values of t

G(t+1)=tG(t)

Question 5

What is the PDF of the gamma distribution

Answer 5

PDF of gamma distribution is:
fx = 1/G(k) * L^k * x^k-1 * e^-Lx for x >=0 L, k > 0
0 otherwise

Question 6

What are mean and variance of gamma distribution

Answer 6

Mean and variance of gamma distribution are:
E(X) = k/L
Var(X)= k/L^2

Question 7

If X~G(L,k) and Y=cX, what is the distribution of Y

Answer 7

Distribution of Y is:
Y~G(L/c ,k)
L is scale

Question 8

What is relation between Gamma and exponential distributions

Answer 8

Relation between Gamma and exponential distributions is:

If x1,…,xk are independent RVs, then x1+…+xk ~Gamma(L,k)

Question 9

If Y1 ~G(L,k1) and Y2~G(L,k2), where k1,k2 are integers, what is distribution of Y1+Y2

Answer 9

Distribution of Y1+Y2 is:

Y1+Y2 ~G(L,k1+k2)

Question 10

What are relations between gamma and normal distributions

Answer 10

Relations between gamma and normal distributions are:
X~N(0,1) then X^2 ~ G(0.5,0.5)
X1,…Xk ~N(0,1) and are independent RVs, then X1+…+Xk ~G(0.5,k/2)