expected value: "Consider 12 independent rolls of a 6-sided die...."

wradwan

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Jan 18, 2017
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2
I have the following question:
Consider 12 independent rolls of a 6-sided die. Let X be the number of 1's and let Y be the number of Ts obtained. Compute E[ X], E[ Y], Var(X), Var(Y), E[X Y], Var(X Y), Coy (X, Y), and p(X, Y). (Hint: You may want to compute the in the order given.)

I answered the following :

E[ X] = np = 12 * 1/6 = 2
E[ Y] = 2 (same as above)
Var(X) = np(1-p) = 5/3
Var(Y) =5/3 (same as above)
E[X + Y] = E(X) + E[Y] = 4
Var(X + Y) = Var(X) + Var(Y) = 10/3 (independent variables)
Cov (X, Y) = E(XY) - E(X+Y) = ? - 4
p(X, Y) No idea
are my answers correct?
 

ksdhart2

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Mar 25, 2016
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Okay, so I'm assuming that the "number of Ts obtained" is actually a typo and you meant "number of 5's obtained." If that's not right, please reply with any necessary corrections. I'll assume it is for the remainder of my post. I agree with all of your answers up through Var(X+Y). Past that, your answers are not correct. For Cov(X,Y), you might find this theorem sheet from Osaka University helpful. In particular, note Theorems 3 and 4:

Theorem [3]: Cov(X, Y) = E(XY) − E(X)E(Y)
and
Theorem [4]: Cov(X, Y) = 0, when X is independent of Y
What does this information suggest the Cov(X,Y) for your problem is? Why? Then for p(X,Y), it looks like you're using a slightly different notation, but if my inference is correct and you're meant to find the correlation coefficient between X and Y, you can also use that sheet from Osaka University to help you out here:

Definition: The correlation coefficient between X and Y, denoted by \(\displaystyle p_{xy}\), is defined as:

\(\displaystyle p_{xy}=\dfrac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)} \cdot \sqrt{\text{Var}(Y)}}=\dfrac{\text{Cov}(X,Y)}{ \sigma_x \cdot \sigma_y}\)
Theorem [6]:\(\displaystyle ρ_{xy} = 0\), when X is independent of Y
What does this information suggest the p(X,Y) for your problem is? Why?
 

wradwan

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Jan 18, 2017
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"is actually a typo and you meant "number of 5's obtained." If that's not right, please reply with any necessary corrections. I'll assume it is for the remainder of my post " My mistake you are right
"
What does this information suggest the Cov(X,Y) for your problem is? Why? " I think it should be 0 since both variables are independent.. Am I right?
 

ksdhart2

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Mar 25, 2016
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964
Sounds right to me. The amount of 1's rolled in no way affects the number of 5's rolled, so the two random variables X and Y must be independent of each other.
 
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