Question

S​uppose continuous random variable X follows exponential distribution with parameter λ=1 i.e. X∼exp (1). Which is the correct answer for conditional probability P[X≥3∣X≥1]?

Answer 1

The correct answer for the conditional probability P[X ≥ 3 | X ≥ 1] when X follows an exponential distribution with parameter λ=1 is e^{-2} due to the memoryless property of the exponential distribution.

Step-by-step explanation:

The subject of this question is conditional probability within the context of continuous random variables and their respective distributions, specifically dealing with the exponential distribution. The exponential distribution is known for its memoryless property, which states that the conditional probability P[X ≥ x + k | X ≥ x] equals P[X ≥ k] regardless of the value of x. Therefore, to find the answer for P[X ≥ 3 | X ≥ 1] when X is exponentially distributed with parameter λ = 1, we can directly use the memoryless property.

P[X ≥ 3 | X ≥ 1] = P[X ≥ 2], because the 'past' event (X being at least 1) does not affect the 'future' probability. Since the exponential distribution with λ = 1 has the probability density function f(x) = e^{-x}, the cumulative distribution function (CDF) for any value 'a' is P(X ≤ a) = 1 - e^{-a}. Thus, P(X ≥ 2) = 1 - P(X ≤ 2) = 1 - (1 - e^{-2}) = e^{-2}.

Consequently, the correct answer for the conditional probability P[X ≥ 3 | X ≥ 1] when X ~ exp(1) is e^{-2}, which is approximately 0.1353.