Question

Let X be an exponential random variable with parameter λ.

a) The probability that X > 5 is

Answer 1

The exponential random variable \( X \) with parameter \( \lambda \) has the cumulative distribution function (CDF) given by:

\[
F_X(x) = 1 - e^{-\lambda x} \quad \text{for } x \geq 0
\]

To find the probability that \( X > 5 \), we can use the complementary cumulative distribution function (CCDF), which is given by:

\[
P(X > x) = 1 - P(X \leq x) = 1 - F_X(x) = e^{-\lambda x}
\]

For \( x = 5 \), the probability is:

\[
P(X > 5) = e^{-\lambda \cdot 5} = e^{-5\lambda}
\]

So, the probability that \( X > 5 \) is:

\[
P(X > 5) = e^{-5\lambda}
\]