Multiple Continuous Random Variables?

A mother has two babies: Jack and Jill. Once she puts them both to sleep, Jack will stay asleep for T hours, where





fT(t) = (K_1)λe^(-λt) for 0 ≤ t ≤ M, 0 otherwise





Jill will stay asleep for S hours, where





fS(s) = K_2μe^(-μt) for 0 ≤ s ≤ M, 0 otherwise





We also assume that S and T are independent.





a.) Find the constants K_1, K_2 in terms of λ, μ, and M





b.) The mother has just put both babies to sleep. What is the probability that she will have a break of at least tq hours before a baby wakes up?





c.) On a good day, jack's sleeping time distribution (given above) has λ = 1. On a bad day, λ = 3. Jack is three times as likely to have a good day than a bad day. Also, the maximum time Jack will sleep on either type of day is 8 hours. On a random day, what is the probability that Jack will sleep 2 hours or more after being put to bed?

Multiple Continuous Random Variables?
a) integral from 0 to M is equal to 1 so integrate w.r.t. t and solve for K_1 and K_2





b) integrate to find F(X) for both babies... 1 - F(X) = P(x %26gt; X) so insert tq in as X and multiply the two together as you assume the babies sleep independent of one another.





c. integrate Jack's function insert the lambdas and then t=2. then since Jack is 3 times as likely to have a good day P(Good) = 3/4 and P(Bad) = 1/4. But remember you have to do it like b) where you do 1-F(X)

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