Let L be the random variable for the length of time, in years, that a person will remember an actuarial statistic. For a certain popula- tion, L is exponentially distributed with mean 1/Y, where Y has a gamma distribution with a = 4.5 and 3 = 4. Find the variance of L.
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No, this solution is not correct. The variance of an exponentially distributed random variable is the square of its mean, so Var(L) = (1/Y)^2. However, the given solution incorrectly multiplies this by Var(Y) to obtain an incorrect expression for Var(L).
The correct approach to find Var(L) involves using the law of total variance. The law of total variance states that Var(L) = E[Var(L|Y)] + Var(E[L|Y]). We already know that Var(L|Y) = (1/Y)^2, so E[Var(L|Y)] = E[(1/Y)^2]. We also know that E[L|Y] = 1/Y, so Var(E[L|Y]) = Var(1/Y).
To find E[(1/Y)^2] and Var(1/Y), we need to use the formulas for the
Once we correctly compute E[(1/Y)^2] and Var(1/Y), we can plug the correct values into the law of total variance to find the correct value for Var(L).
by Bartleby ExpertBut isn't the
by Bartleby Expertbut isn't the expectation alpha times beta rather than alpha divided by beta?
by Bartleby Expert