Question

Let \( X_1, X_2, X_3, \dots \) be independent and identically distributed random variables with \( E[X_1] = \mu \). Let \( N \) be a positive integer valued random variable with \( E[N] = n \). If \( S_N = X_1 + X_2 + \dots + X_N \), then \( E[S_N] = \dots \)

• A. \( \mu \)
• B. \( N \mu \)
• C. \( n \mu \)
• D. \( \mu n \)

Hint:

When dealing with sums of random variables, use the linearity of expectation: \( E[S_N] = E[N] \cdot E[X] \).

Answer 1

The Correct Option is C

Step 1: Understand the setup of the problem 
We are given that \( X_1, X_2, X_3, \dots \) are independent and identically distributed (i.i.d.) random variables, and \( E[X_1] = \mu \). Additionally, we are told that \( N \) is a positive integer-valued random variable, with \( E[N] = n \). The sum of these \( N \) random variables is defined as: \[ S_N = X_1 + X_2 + \dots + X_N \] Our goal is to find the expectation \( E[S_N] \). 
Step 2: Use the linearity of expectation 
The expectation of the sum of random variables is the sum of the expectations of those random variables. Thus: \[ E[S_N] = E[X_1 + X_2 + \dots + X_N] \] By the linearity of expectation, this is: \[ E[S_N] = E[X_1] + E[X_2] + \dots + E[X_N] \] Since each \( X_i \) has the same expectation \( \mu \), we have: \[ E[S_N] = N \cdot \mu \] Step 3: Consider the expectation of \( N \) 
Since \( N \) is a random variable itself, we must account for its expectation. Thus: \[ E[S_N] = E[N] \cdot \mu = n \cdot \mu \] Conclusion: The expected value of \( S_N \) is \( n \mu \), where \( n \) is the expected value of \( N \).