Question

Suppose a continuous random variable X follows an exponential distribution with parameter λ = 1. Which is the correct answer for conditional probability P(X ≥ 3 | X ≥ 1)?

a) 1/2
b) 1/3
c) 2/3
d) 1

Answer 1

The conditional probability P(X ≥ 3 | X ≥ 1) is approximately 0.6665 when a continuous random variable X follows an exponential distribution with parameter λ = 1.

Step-by-step explanation:

The exponential distribution is related to the Poisson distribution. If a continuous random variable X follows an exponential distribution with parameter λ = 1, then the number of events per unit time follows a Poisson distribution with mean λ = 1/μ. The conditional probability P(X ≥ 3 | X ≥ 1) can be calculated using the properties of the exponential distribution.

Using the cumulative distribution function, P(X < x) = 1 - e(−λ)(x), we can calculate P(X < 1) = 1 - e(−1)(1) ≈ 0.3935 and P(X < 3) = 1 - e(−1)(3) ≈ 0.9502. Therefore, P(X ≥ 3 | X ≥ 1) = (P(X ≥ 3) - P(X < 1)) / (1 - P(X < 1)) = (1 - 0.9502) / (1 - 0.3935) ≈ 0.6665.