Question

Suppose that X is an exponentially distributed random variable with λ = 0.37

Find each of the following probabilities:
A. P(X > 1)
B. P(X > 0.29)
C. P(X < 0.37)
D. P(0.31 < X < 2.04)

Answer 1

A. P(X > 1) = e^{-0.37 * 1} ≈ 0.692

B. P(X > 0.29) = e^{-0.37 * 0.29} ≈ 0.805

C. P(X < 0.37) = 1 - P(X ≥ 0.37) = 1 - e^{-0.37 * 0.37} ≈ 0.289

D. P(0.31 < X < 2.04) = P(X < 2.04) - P(X < 0.31) = e^{-0.37 * 2.04} - e^{-0.37 * 0.31} ≈ 0.186

Step-by-step explanation:

An exponentially distributed random variable X with a rate parameter (λ) follows the probability density function f(x) = λe^-λx for x ≥ 0. In this case, λ is given as 0.37.

For part A, finding P(X > 1) involves integrating the probability density function from 1 to infinity. The formula for this is P(X > 1) = ∫₁^∞ λe^-λxdx, which simplifies to e^{-0.37 * 1} ≈ 0.692.

For part B, calculating P(X > 0.29) requires integrating the probability density function from 0.29 to infinity. The formula simplifies to e^{-0.37 * 0.29} ≈ 0.805.

For part C, P(X < 0.37) is found by subtracting P(X ≥ 0.37) from 1. The calculation is 1 - e^{-0.37 * 0.37} ≈ 0.289.

For part D, finding P(0.31 < X < 2.04) involves subtracting P(X < 0.31) from P(X < 2.04). The calculation is e^{-0.37 * 2.04} - e^{-0.37 * 0.31} ≈ 0.186. These calculations represent standard procedures for working with exponential distributions and probability density functions.