# Thread: Variance and mean of profit

1. ## Variance and mean of profit

Hi, I'm having a little trouble solving this problem. I understand how to solve it for tossing a die because there are values associated it. In this case I get to draw 5 cards from a deck (then immediately replace it) and I receive \$4 for every heart drawn. I have to find the mean and variance of my payoff.

I understand the the probablity of drawing a heart is 13/52 but I'm not sure how to calculate the variance and mean of the profit using this information. I think this is a simpler problem than I'm making it but I'm just a little stuck and I'd appreciate any help. Thanks!

2. Thats a pretty good game since you didn't mention any fee for playing it...

Hint Your profit is 4X, where X~Bi(5, [13/52])

Can you calculate the Mean, variance of X?
And hence can you find the Mean, Variance of 4X?

If you haven't been taught how to do the above, then you can alternatively calculate the probability distribution of profit:
tips on how to do that are in the spoiler, but using the hint above is faster and easier.
Spoiler:

P(profit=0) = P(draw 0 hearts)
P(profit=4) = P(draw 1 heart)
P(profit=8) = P(draw 2 hearts)
....

Once you have the probability distribution of the profit, you can calculate the mean and variance just like for any other probability distribution.

3. Thanks so much for your reply. I've tried doing it a few different ways and it doesn't seem to be working out as I'd like.

So I calculate the mean as:
E(X)=13/52*(0+4+8+12+16+20)=15
E(4X)=4*E(X) = 60

And the variance as:
var(X)=(13/52)[{0-15}^{2}+ {4-15}^{2}+{8-15}^{2}+{12-15}^{2}+{16-15}^{2}+{20-15}^{2}] =107.5 but this number does not seem correct at all
var(4X) = 4*var(X) = 430

I know these aren't the correct numbers but I'm not sure where I'm going wrong in my calculations since these are the correct formulas (I think). Can anyone point out what I'm doing wrong here?

4. There are numerous errors im afraid.

E(X)=13/52*(0+4+8+12+16+20)=15
You seem to have defined X as your profit (this is fine)
You seem to have assumed that P(X=0) = P(X=4) = P(X=8) ... =P(X=20) = 13/52. This is not correct. Use the binomial distribution to get the probabilities. Note that the profit is not binomially distributed, but the number of hearts drawn is.

E(4X)=4*E(X) = 60
You already defined X = profit, so you aren't interested in 4X. (Before you say i told you to do that, this is not what i meant).

var(X)=(13/52)[{0-15}^{2}+ {4-15}^{2}+{8-15}^{2}+{12-15}^{2}+{16-15}^{2}+{20-15}^{2}] =107.5
The constant probability assumption is still incorrect, and your E(X) error is carried forward from the previous calculations.

var(4X) = 4*var(X) = 430
Var(4X)=4^2 Var(X) = 16 Var(X)

However, because you defined X as profit, you aren't interested in Var(4X). The answer is Var(X).

Here is one possible solution. I assume you have been taught the properties of the binomial distribution and can use them without derivation in your answer
Define X = Number of Hearts drawn
X ~ Bi(5,(13/52))

The variance and expectation of a biniomial distribution are well known:

E(X) = 5 * 13/52 = 1.25
Var(X) = 5 * (13/52) * (39/52) = 15/16

Define Y = Profit. Note that Y = 4X
E(Y) =E(4X) = 4E(X) = 1.25*5 = 6.25
Var(Y) = Var(4X) = 16 Var(X) = 16*15/16 = 15

5. Oh I see now, I didn't define things the right way to begin with. I very much appreciate your help and taking the time to explain!