# Math Help - Poisson distribution problem help

1. ## Poisson distribution problem help

The number of cases , $X$, arriving at hospital's has a Poisson distribution with rate of 16 per night. Doctor's basic shift is 12 hours. Each case on shift will take an extra $1/8$ of an hour. Assume that a single doctor sees all cases.

Part A: Let $Y$ represent the length of a doctor's shift. Write an equation for $Y$ in terms of $X$.

My answer is : $Y=12+X/8$. I think it is correct.

Part B: Calculate expected value and variance of Y.
This part I'm totally lost. What should I do?

2. Originally Posted by kleenex
The number of cases , $X$, arriving at hospital's has a Poisson distribution with rate of 16 per night. Doctor's basic shift is 12 hours. Each case on shift will take an extra $1/8$ of an hour. Assume that a single doctor sees all cases.

Part A: Let $Y$ represent the length of a doctor's shift. Write an equation for $Y$ in terms of $X$.

My answer is : $Y=12+X/8$. I think it is correct.

Part B: Calculate expected value and variance of Y.
This part I'm totally lost. What should I do?

For any random variable X:

E(aX + b) = aE(X) + b.

Var(aX + b) = $a^2$ Var(X).

And you know E(X) and Var(X), right?

3. Originally Posted by mr fantastic
For any random variable X:

E(aX + b) = aE(X) + b.

Var(aX + b) = $a^2$ Var(X).

And you know E(X) and Var(X), right?
$E(aX + b) = E(X/8 + 12) = 1/8 E(X) +12$
$-12 = 1/8 E(X)$
$E(X) = -96$

I think I misinterpret what you said, could you give a bit more explain. Thank you

4. Originally Posted by kleenex
If I understand your reply right, then I will have $E(aX + b) = E(X/8 + 12) = 1/8 E(X) +12$
$-12 = 1/8 E(X)$
$E(X) = -96$

I think I misinterpret what you said, could you give a bit more explain. Thank you
I'm afraid you have completely misunderstood. Let me be crystal clear: Y = aX + b.

Is it clear to you what the random variables X and Y represent in your problem? It should be because the question defines them.

Is it clear to you what you're trying to find? You're trying to solve for E(Y) and Var(Y).

Is it clear to you that you're essentially given the value of E(X) and Var(X) ....?

5. Ok, I'm clear now. Thank you