Let X and Y be zero-mean independent random variables. Which one of the following statements is correct? Hint: You can take for granted the intuitive fact that E[X/X=x] = x
a) E[X+Y/X] = 0
b) E[X+Y/X] = x
c) E[X+Y/X] = X
d) E[X+Y/X] = X+Y
c) E[X+Y/X] = X
Why?
To find the conditional expectation E[X+Y/X], we condition on the value of X. Since X is a zero-mean random variable, when we condition on X, it takes on a fixed value x.
So, we can write E[X+Y/X] as E[X+Y | X=x].
By linearity of expectation, we can split this into E[X | X=x] + E[Y | X=x].
Since E[X/X=x] = x, we have E[X | X=x] = x.
Since X and Y are independent, E[Y | X=x] = E[Y] = 0, since Y is also a zero-mean random variable.
Therefore, E[X+Y | X=x] = x+0 = x.
So, the correct statement is c) E[X+Y/X] = X.
To find the correct statement, we can simplify the expression E[X+Y/X] using the linearity of expectation.
E[X+Y/X] = E[X/X] + E[Y/X]
Since X is a zero-mean random variable, E[X/X] = E[1] = 1.
E[X+Y/X] = 1 + E[Y/X]
Since X is a zero-mean random variable, E[Y/X] = 0.
E[X+Y/X] = 1 + 0 = 1
Therefore, the correct statement is:
b) E[X+Y/X] = x
To find the correct statement, let's use the properties of expected value and the fact that X and Y are independent random variables.
Recall that the expected value of a random variable is a measure of its central tendency. Specifically, for a random variable X, E[X] represents the average value of X over all possible outcomes.
Now let's consider each option:
a) E[X+Y/X] = 0
This option implies that the expected value of the sum of X and Y, divided by X, is equal to 0. However, we know that X and Y are independent random variables with zero mean. Since the expected value of X is 0, the expected value of X+Y cannot be 0 unless Y is always 0. Therefore, option a) is incorrect.
b) E[X+Y/X] = x
This option states that the expected value of the sum of X and Y, divided by X, is equal to the value of X. While it may seem plausible, it is not correct. To see why, let's calculate E[X+Y/X]:
E[X+Y/X] = E[X/X] + E[Y/X]
= x + E[Y/X]
The value of E[Y/X] depends on the values of X and Y and their joint distribution. So, in general, E[Y/X] cannot be simplified further to just x. Therefore, option b) is incorrect.
c) E[X+Y/X] = X
This option suggests that the expected value of the sum of X and Y, divided by X, is equal to the random variable X itself. To check this option, let's calculate E[X+Y/X]:
E[X+Y/X] = E[X/X] + E[Y/X]
= x + E[Y/X]
Again, the value of E[Y/X] depends on the values of X and Y and their joint distribution. However, we do know that E[X/X] = x. Therefore, E[X+Y/X] simplifies to:
E[X+Y/X] = x + E[Y/X]
Since x + E[Y/X] is a combination of a constant (x) and a random variable (E[Y/X]), it cannot be simplified further to just X. Therefore, option c) is incorrect.
d) E[X+Y/X] = X+Y
This option states that the expected value of the sum of X and Y, divided by X, is equal to the sum of X and Y. Let's calculate E[X+Y/X]:
E[X+Y/X] = E[X/X] + E[Y/X]
= x + E[Y/X]
Since E[Y/X] is a combination of X and Y (dependent on their joint distribution), we cannot simplify it further to just Y. Therefore, E[X+Y/X] is equal to X + E[Y/X], which is not equal to X + Y. Therefore, option d) is incorrect.
None of the given options are correct for the statement.