Question

Suppose that f(x) = e-x for x > 0. Determine the following probabilities:

Round your answers to 4 decimal places.
P(X > 1) =________.

Answer 1

For the exponential function f(x) = e^-x, the probability that X is greater than 1, represented as P(X > 1), can be calculated using the formula for the exponential distribution, resulting in approximately 0.3679 when rounded to four decimal places.

Step-by-step explanation:

Given the function f(x) = e-x for x > 0, we use the properties of the exponential distribution to calculate probabilities. To determine P(X > 1), we can utilize the cumulative distribution function (CDF) formula for the exponential distribution, which is P(X < x) = 1 - e-mx. Using this, we can derive that P(X > x) = e-mx since P(X < x) = 1 - P(X > x).

Therefore, for our specific function where m = 1 and for x > 1, we have:

P(X > 1) = e-1 ≈ 0.3679

This result is obtained by substituting x with 1 in the derived formula P(X > x) = e-mx. Hence, the probability that X is greater than 1 is approximately 0.3679 when rounded to four decimal places.