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Question 1, About Exponential Distribution Exponential distribution is a very important distribution in this course. We will use it frequently in our future lectures. Suppose a non-negative real valued random variable X obeys an exponential distribution with parameter ?. That is, the probability density function of X is f(X = x) = ?e^{-?x}, x ? 0. a) Prove that X has the memoryless property. That is, the p.d.f. f(X = x + t|X > t), x > x_0, also has the same form as f(X = x). b) Calculate the coefficient of variability of X, C^2{X}, where C^2{X} := Var{X}/(E{X})^2. (write the detailed calculation process) Note: please use Laplace transform. c) For the two independently exponentially distributed random variables X_1 and X_2 with parameter ?_1 and ?_2, respectively, calculate the probability P(X_1 < X_2).

          Question 1, About Exponential Distribution

Exponential distribution is a very important distribution in this course. We will use it frequently in our future lectures. Suppose a non-negative real valued random variable X obeys an exponential distribution with parameter ?. That is, the probability density function of X is f(X = x) = ?e^{-?x}, x ? 0.

a) Prove that X has the memoryless property. That is, the p.d.f. f(X = x + t|X > t), x > x_0, also has the same form as f(X = x).

b) Calculate the coefficient of variability of X, C^2{X}, where C^2{X} := Var{X}/(E{X})^2. (write the detailed calculation process) Note: please use Laplace transform.

c) For the two independently exponentially distributed random variables X_1 and X_2 with parameter ?_1 and ?_2, respectively, calculate the probability P(X_1 < X_2).

Question 1, About Exponential Distribution

Exponential distribution is a very important distribution in this course. We will use it frequently in our future lectures. Suppose a non-negative real valued random variable X obeys an exponential distribution with parameter ?. That is, the probability density function of X is f(X = x) = ?e^-?x, x ? 0.

a) Prove that X has the memoryless property. That is, the p.d.f. f(X = x + t|X > t), x > x0, also has the same form as f(X = x).

b) Calculate the coefficient of variability of X, C^2X, where C^2X := VarX/(EX)^2. (write the detailed calculation process) Note: please use Laplace transform.

c) For the two independently exponentially distributed random variables X1 and X2 with parameter ?1 and ?2, respectively, calculate the probability P(X1 < X2).

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