Can E(Z) be determined from E(X) and E(Y) when X and Y are independent?

In summary, the conversation discusses the problem of finding the expected value of variable Z, which is dependent on variables X and Y, given that X and Y are independent with known expected values. The conversation also mentions the need to know the variance of X in order to determine the expected value of Z.
  • #1
oyth94
33
0
Hi I know this may be a silly question but i am doubting myself on how i did this question:

Suppose X and Y are independent, with E(X) = 5 and E(Y) = 6. For each of the following variables Z, either compute E(Z) or explain why we cannot determine
E(Z) from the available information:
Z = (2+X)(3X + 4Y)= 6X + 8Y + 3X^2 + 4XY

So I did E(Z= 6X + 8Y+ 3X^2 + 4XY) = 6E(X) + 8E(Y) + 3E(X^2) + 4E(X)(Y)
Im not sure if i am on the right track so far (ie i just have to plug in E(X) and E(Y) to find E(Z)
but doesn't this require some integrals since it is in the absolute continuous case? Otherwise if we plug it in isn't it not independent because the answer doesn't equal to E(XY)?
 
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  • #2
Re: expected value independence

The problem is that You know $E\{X\}$ and $E\{Y\}$, that X and Y are independent and nothing else. That permits You to find $E\{X Y\}$ but for $E\{X^{2}\}$ You have to know $\text{Var} \{X\}$ and apply the identity $\text{Var} \{X\} = E \{X^{2}\} - E^{2} \{X\}$... Kind regards $\chi$ $\sigma$
 

FAQ: Can E(Z) be determined from E(X) and E(Y) when X and Y are independent?

What is expected value independence?

Expected value independence is a statistical concept that refers to the idea that the expected value of one variable is not affected by the value of another variable. In other words, the two variables are independent of each other.

Why is expected value independence important in statistics?

Expected value independence is important because it allows us to make more accurate predictions and draw more reliable conclusions from our data. It also helps us to identify and understand relationships between variables in a dataset.

How is expected value independence calculated?

The expected value of a variable is calculated by multiplying each possible value of the variable by its probability of occurring, and then summing up all of these products. If the expected value of one variable is not affected by the value of another variable, then they are considered to be independent.

What are some real-world examples of expected value independence?

One example of expected value independence is in gambling. In games like roulette, the expected value of a bet on a single number is always the same, regardless of previous outcomes. Another example is in insurance, where the expected value of a policy does not change based on the age or gender of the policyholder.

What are the limitations of expected value independence?

Expected value independence assumes that the variables being analyzed are truly independent and do not influence each other in any way. This may not always be the case, and in some situations, the expected values may be affected by hidden or unknown factors.

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