Question

Let $X$ be an exponential random variable with mean parameter one. Then the conditional probability $P(X>10 | X>5)$ is equal to

• A. $1 - e^{-5/2}$
• B. $\dfrac{5}{e^2}$
• C. $1 - e^{-5}$
• D. $e^{-5}$

Hint:

The exponential distribution is memoryless: $P(X>a + b | X>a) = P(X>b)$.

Answer 1

The Correct Option is C

Step 1: Exponential Distribution Memoryless Property
For an exponential distribution with $\lambda = 1$, we have: \[ P(X>a + b | X>a) = P(X>b) \] Step 2: Apply memoryless property
\[ P(X>10 | X>5) = P(X>5) = 1 - P(X \leq 5) = 1 - (1 - e^{-5}) = e^{-5} \] So the correct answer is: \[ \boxed{e^{-5}} = \boxed{1 - (1 - e^{-5})} = 1 - e^{-5} \]