How Do Conditional Probabilities and Expectations Relate in Statistics?

[MaxManus]

Homework Statement

Consider two random variables X and Y. Suppose that Y takes on k values y_i,...,y_k and X takes l values x_i,...,x_l

Hey, I am supposed to show that
1) Pr(Y = y_i = $$\sum$$,from 1 to l Pr(Y = y_i(givrn)(X=x_i)Pr(X=x_i)


and
2) E(Y) = E(E(Y(given)X)


I am not sure where to start on neither of them,

 

[quantumdude]

You need to learn how to do a little research, bro. That second problem is worked out completely on a wikipedia page, which I was able to find in about 20 seconds. I will let you think about what keywords to search for (you should at least know what E(Y) is called). As for the first problem, I'm guessing that you didn't type it in correctly because I can't parse what you're trying to say. Try again?

 

[Architeuthis Dux]

can you please help me?



Sure, let's break down the two parts separately.

1) The first part is asking you to show that the probability of Y taking on the value yi is equal to the sum of the probabilities of Y taking on the value yi for each possible value of X. In other words, the probability of Y = yi is the sum of the probabilities of Y = yi given each possible value of X. This can be represented as:

Pr(Y = yi) = ∑ Pr(Y = yi | X = xi) * Pr(X = xi)

To prove this, you can use the definition of conditional probability: Pr(A|B) = Pr(A and B) / Pr(B). Substituting this into the equation above, we get:

Pr(Y = yi) = ∑ Pr(Y = yi and X = xi) / Pr(X = xi)

Now, we can use the fact that Y and X are independent random variables to simplify the equation. This means that the probability of Y taking on a certain value is not affected by the value of X. Therefore, Pr(Y = yi and X = xi) = Pr(Y = yi) * Pr(X = xi). Substituting this into the equation above, we get:

Pr(Y = yi) = ∑ Pr(Y = yi) * Pr(X = xi) / Pr(X = xi)

The Pr(X = xi) terms cancel out, leaving us with:

Pr(Y = yi) = ∑ Pr(Y = yi)

which proves the statement.

2) The second part is asking you to show that the expected value of Y is equal to the expected value of the conditional expected value of Y given X. This can be represented as:

E(Y) = E(E(Y | X))

To prove this, we can use the definition of expected value: E(X) = ∑ x * Pr(X = x). Substituting this into the equation above, we get:

E(Y) = ∑ y * Pr(Y = y)

Now, we can use the definition of conditional expected value: E(X | Y) = ∑ x * Pr(X = x | Y = y). Substituting this into the equation above, we get:

E(Y) = ∑ ∑ y * Pr(X = x | Y = y) * Pr(Y = y)

We can rearrange the terms to get:

E(Y) = ∑ ∑ y