Question

It can be shown that the sum of independent exponential random variables with mean β has a Gamma distribution, and if there are α exponentials in the sum, then the Gamma distribution has parameters α and β.

Answer 1

The probability that a randomly selected student spends at most 125 hours on the project is approximately 0.999972.

Step-by-step explanation:

The time spent by students on a particular project has a gamma distribution with parameters α=50 and β=2. To compute the probability that a randomly selected student spends at most 125 hours on the project, we can use the cumulative distribution function (CDF) of the gamma distribution.

First, we need to convert the hours spent from gamma distribution to a gamma distribution with parameters α=1 and β=1. This can be done by dividing the hours spent by β, so the new parameters are α=50 and β=125/2=62.5.

Next, we can use the CDF of the gamma distribution to calculate the probability. Plugging in the values α=50, β=62.5, and x=125 into the CDF formula, we get P(X ≤ 125) = 0.999972.

Complete Question:

It can be shown that the sum of independent exponential random variables with mean β has Gamma distribution and if there are α exponentials in the sum then Gamma distribution has parameters α and β. Suppose that the time X (in hours) spent by students on a particular project has gamma distribution with parameters α=50 and β=2. Compute the approximate probability that randomly selected student spends at most 125 hours on that project.