Conditional exponential probability

[Longines]

Hello all,

I've been stuck on this question for a while and it's annoying the stew out of me!

I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help?

Question:
Let X be a random variable and A be an event such that, conditional on A, X is exponential with parameter λ, and conditional on $A^c$ (A complement), X is exponential with parameter μ.
Write E[X] in terms of λ, μ and p, the probability of A

 

[lfdahl]

Hello

Thankyou for sharing your problem with us at the MHB! :)

I suggest, you make a first step by writing down the law of total expectation:

$E[X] = \sum_{i=1}^{n} E[X|A_i]\cdot P(A_i)$

In this specific case the sum has only two terms ($n=2$)

 

[Longines]

Hello

Thankyou for sharing your problem with us at the MHB! :)

I suggest, you make a first step by writing down the law of total expectation:

$E[X] = \sum_{i=1}^{n} E[X|A_i]\cdot P(A_i)$

In this specific case the sum has only two terms ($n=2$)

I figured it had something to do with this, but I don't understand how I express it with lambda and such.

Could you please elaborate?

 

[I like Serena]

I figured it had something to do with this, but I don't understand how I express it with lambda and such.

Could you please elaborate?

The expectation of an exponential distribution with parameter $\lambda$ is $\frac 1 \lambda$.
So:
$$E[X|A] = \frac 1 \lambda$$

 

[lfdahl]

I figured it had something to do with this, but I don't understand how I express it with lambda and such.

Could you please elaborate?

First step:$E[X] = E[X|A] \cdot P(A)+E[X|A^c] \cdot P(A^c)$According to #4 you know $E[X|A]$ and $E[X|A^c]$ as $\frac{1}{\lambda}$ and $\frac{1}{\mu}$ respectively.If $P(A)=p$ then what is $P(A^c)$?

Now, try to express $E[X]$ in terms of $\lambda$, $\mu$ and $p$.

 

[chisigma]

Hello all,

I've been stuck on this question for a while and it's annoying the hell out of me!

I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help?

Question:
Let X be a random variable and A be an event such that, conditional on A, X is exponential with parameter λ, and conditional on $A^c$ (A complement), X is exponential with parameter μ.
Write E[X] in terms of λ, μ and p, the probability of A

If $P \{A\} = p$ and $P \{A^{c}\} = 1 - p$, then is...

$\displaystyle E \{X\} = \frac{p}{\lambda} + \frac{1 - p}{\mu}\ (1)$

Kind regards

$\chi$ $\sigma$