Question

While Juanita waits for her Mac and Cheese to boil, she works on her homework for 1/2 of the time. She has been waiting for 3 minutes already (so she has already done 1.5 minutes of homework). She doesn’t know how much more time is needed. She estimates that the remaining time (in minutes) until the water boils is an Exponential random variable Y with density f(Y) (y) = 1/4e^-y/4 for y > 0, and fY (y) = 0 otherwise. Find the total
length of time that she expects to work on her homework while waiting for the
water to boil.

y=4 for y > 0, and fY (y) = 0 otherwise. Find the total length of time that she expects to work on her homework while waiting for the water to boil.

Answer 1

Let Y denote the amount of remaining time she waits for the water to boil, then given that

Y ~ Exp(1/4) Distribution.

Then out of the Y amount of minutes she will spend Y/2 doing the homework, so total time spend doing the homework,

Z = 3/2 + Y/2 -- as she has aslready spent 3/2 time doing her homework and out of the remaining time she we spend Y/2 again on homework so total time = 3/2 + Y/2

So E(Z) = 3/2 + E(Y/2)

Now since Y ~ Exp(1/4) implies E(Y) = 4. --- Result Expectation of exponential distribution with parameter lambda = 1/lambda

So E(Z) = 3/2 + 4/2 = 7/2 = 3.5

So she works on her homework for 3.5 minutes on average before the water boils