Question

Show that if a random variable X has an exponential distribution, then for t > 0 and x > 0, PX > t | X > x = PX > t - x.

a. True
b. False

Answer 1

To show that if a random variable X has an exponential distribution, then for t > 0 and x > 0, PX > t | X > x = PX > t - x is true. This can be proven using the memoryless property of the exponential distribution.

Step-by-step explanation:

To show that if a random variable X has an exponential distribution, then for t > 0 and x > 0, PX > t | X > x = PX > t - x is true, we can use the memoryless property of the exponential distribution. The memoryless property states that P(X > x + k|X > x) = P(X > k). In this case, we have P(X > t| X > x) = P(X > t - x). Therefore, the given statement is true. This inference highlights that the probability of X exceeding t given that X is greater than x is equivalent to the probability of X surpassing t−x. This relationship, grounded in the memoryless property, confirms the validity of the provided statement within the context of the exponential distribution.