Question

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

Round your answers to 4 decimal places.
P(X>1) = 0.3679
P(1P(X=3) = ???

Answer 1

The probability P(X = 3) for a continuous distribution like f(x) = e^(-x) is zero. Other probabilities such as P(1 < x < 4) and P(x ≥ 8) would be calculated using the cumulative distribution function, provided a rate parameter is given.

Step-by-step explanation:

Given f(x) = e^(-x) for x > 0, this is a continuous distribution, and therefore the probability at any single point, such as P(X = 3), is zero. Instead, we can find probabilities over intervals. For P(1 < x < 4), you would calculate the difference between the cumulative distribution function (CDF) values at 4 and 1. To find P(x ≥ 8), you would calculate the complementary probability of P(X < 8) using the CDF.

Unfortunately, without a specific rate parameter (λ) for the exponential distribution, we can't provide numerical answers. If a parameter were given, such as λ = 0.5, you could use the CDF P(X < x) = 1 − e^(-λx) to find the probabilities for each scenario by substituting λ and x with appropriate values and solving accordingly.