9

Question 1

X is an exponential random variable of parameter lambda when its probability distribution function is: f(x) = ?

Answer 1

{lambda e^-(lambda x), x >= 0
{0, x < 0

Question 2

Expression for the cumulative function of exponential random variable of parameter lambda:

Answer 2

1 - e^-(lambda a)

Question 3

P{X < a} in exponential random variable?

Answer 3

1 - e^-(lambda a)

Question 4

P{X > a} in exponential random variable:

Answer 4

e^-(lambda a)

Question 5

Expected value of X^n in exponential random variable: E[X^n] = ?

Answer 5

-int(0, oo)(x^n lambda e^-(lambda x) dx) = (n/lambda) E[X^(n-1)]

Question 6

Expectation of exponential random variable: E[X] = ?

Answer 6

1/lambda

Question 7

Variance in exponential random variable: Var[X] = ?

Answer 7

(2/lambda^2) - 1/lambda^2 = 1/lambda^2

Question 8

If X1 and X2 are independent and exponential with parameters lambda1 and lambda 2, then X = min{X1, X2} is exponential with parameter?

Answer 8

lambda = lambda1 + lambda2. Since X1 and X2 are independent, P{X > a} = P{X1 > a}P{X2 > a} = e^-(lambda1 a) e^-(lambda2 a) = e^-(lambda a)

Question 9

Memoryless property?

Answer 9

The time is independent on the probability of the next event.