Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I randomly chose either Alice or Bob to go catch a penguin for me with equal probabilities of chosing either person. Let $I=0$ if I chose Alice and $I=1$ if I chose Bob. Alice can catch a penguin in time $T_1 \sim Exponential(\lambda_1)$. Bob can catch a penguin in time $T_2 \sim Exponential(\lambda_2)$. Let $T$ be the time it takes for a penguin to be caught. What is the variance of $T$?

I started this problem with:

$$ Var(T) = E(Var(T|I)) + Var(E(T|I)) $$

However, I'm not sure of how to calculate either $E(Var(T|I))$ or $Var(E(T|I))$.

share|improve this question
1  
This looks a lot like homework, and if so, please add the homework tag. With regard to the question itself, $\text{var}(T\mid I)$ is a random variable that takes on two values depending on whether $I$ is $0$ or $1$. What is its average value? Similarly, $E[T\mid I]$ is a random variable that takes on two values. What is its average? What is its variance? You will get a lot farther if you write down the two values mentioned above (in the two cases) explicitly and then proceed, instead of trying to deal with mystical magical formulas in generality. –  Dilip Sarwate Dec 13 '12 at 21:00
add comment

1 Answer 1

up vote 4 down vote accepted

HINT: Start by computing $E(T|I)$ and $Var(T|I)$, these are just the expectation and variance of an exponentialy distributed variable thus

$$E(T|I)=\frac{1}{\lambda_{I+1}} \text{ and } Var(T|I)=\frac{1}{(\lambda_{I+1})^2} \; .$$

Can you take it from here?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.