Expected value of a random variable defined by the sum of other N random variables

Question

could anyone help me understand better this problem, please?

$Y$ is a random variable that is defined by another random variable $N$ and a sequence of random variables $X_1$, $X_2$,$\dots$, such that:

$$ Y = \sum_{i=1}^NX_i$$

Obs: $X_1,X_2,X_3,\dots$ is a sequence of independent and identically distributed random variables and $N$ is a non negative random variable, which values are in the set of positive integers and independent of the sequence $\{X_i: i ≥ 1\}$.

Question: What is the expected value of $Y$ in terms of $\mathbb E\left[X_1\right]$ (Expected value of $X_1$) and $\mathbb E[N]$ (Expected value of $N$)

My doubt: If $Y$ was defined by only one random variable, I would use the law of total probability, but here $Y$ is defined by two random variables! Could anyone give me a direction without solving the problem, please?

Obs2: I'm in first semester probability course, so I might not understand too advanced topics.

Answer 1

To find the expected value of $Y$ in terms of $E[X_1]$ and $E[N]$, we can use the law of total expectation. The law of total expectation states that for any random variables $X$ and $Y$:

$$E[Y] = E[E[Y|X]]$$

Applying this to the given problem, we can express the expected value of $Y$ as:

$$E[Y] = E[E[Y|N]]$$

Now, let's focus on finding $E[Y|N]$. Given a fixed value of $N$, we can rewrite $Y$ as:

$$Y = X_1 + X_2 + \dots + X_N$$

Since $X_1, X_2, \dots$ are independent and identically distributed random variables, their expected value is the same. Let's denote it as $\mu = E[X_1]$. Using this, we can express the expected value of $Y$ given a fixed value of $N$ as:

$$E[Y|N] = E[X_1 + X_2 + \dots + X_N | N]$$ $$= E[X_1 | N] + E[X_2 | N] + \dots + E[X_N | N]$$ $$= N \cdot E[X_1 | N]$$

Now, we can substitute this back into the expression for $E[Y]$:

$$E[Y] = E[E[Y|N]]$$ $$= E[N \cdot E[X_1 | N]]$$

At this point, we need to consider the relationship between $N$ and $X_1$ in order to proceed further. Without additional information about their relationship or the distributions involved, it is not possible to provide a more specific expression for $E[Y]$ in terms of $E[X_1]$ and $E[N]$.

However, if $N$ and $X_1$ are independent, we can simplify the expression further:

$$E[Y] = E[N \cdot E[X_1 | N]]$$ $$= E[N] \cdot E[X_1]$$

This simplification is possible because if $N$ and $X_1$ are independent, then $E[X_1 | N] = E[X_1]$.