Question

If \(X\) is a nonnegative integer valued random variable, show that (a) $$ E[X]=\sum_{n=1}^{\infty} P[X \geq n\\}=\sum_{n=0}^{\infty} P(X>n\\} $$ Hint: Define the sequence of random variables \(I_{n}, n \geq 1\), by $$ I_{n}=\left\\{\begin{array}{ll} 1, & \text { if } n \leq X \\ 0, & \text { if } n>X \end{array}\right. $$ Now express \(X\) in terms of the \(I_{n}\). (b) If \(X\) and \(Y\) are both nonnegative integer valued random variables, show that $$ E[X Y]=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m) $$

Short answer

The short answer based on the above step-by-step solution is: (a) To prove \(E[X] = \sum_{n=1}^{\infty} P(X \geq n)\), we first express \(X\) as a sum of the variables \(I_n\), where \(I_n = 1\) if \(n \leq X\) and 0 otherwise. Then, we find the expected value of each \(I_n\) with \(E[I_n] = \sum_{n=1}^{\infty} P(X \geq n)\). Finally, we compute the expected value of \(X\) as the summation of the expected values of \(I_n\) to obtain \(E[X] = \sum_{n=1}^{\infty} P(X \geq n) = \sum_{n=0}^{\infty} P(X > n)\). (b) To prove \(E[XY] = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m)\), we first express \(X\) and \(Y\) in terms of \(I_n\) and \(J_m\) sequences, respectively. Then, we rewrite the product \(XY\) as the double sum of the product of \(I_n\) and \(J_m\). Finally, we compute the expected value of \(XY\) by finding the expected value of the product \(I_n J_m\), which gives us the desired result: \(E[XY] = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m)\).

Step-by-step solution

Express X in terms of I_n

First, let's express \(X\) in terms of the variables \(I_n\). Recall the definition of \(I_n\): $$ I_{n}=\left\\{\begin{array}{ll} 1, & \text { if } n \leq X \\\ 0, & \text { if } n>X \end{array}\right. $$ Now, the sum of all \(I_n\) from 1 to \(X\) is equal to the value of X. This is because \(I_n\) is 1 for all \(n \leq X\) and 0 for \(n > X\). Therefore, $$ X = \sum_{n=1}^{\infty} I_n $$

Finding E[I_n]

To find the expected value of \(X\), we first need to find the expected value of each \(I_n\). Let's find \(E[I_n]\) using the definition of the expected value: $$ E[I_n] = \sum_{n=1}^{\infty} I_n P(I_n = 1) $$ Using the definition of \(I_n\) (and noticing that \(I_n\) is either 0 or 1), we can rewrite the expected value as: $$ E[I_n] = \sum_{n=1}^{\infty} P(I_n = 1) = \sum_{n=1}^{\infty} P(X \geq n) $$

Finding E[X] for part (a)

Using the expression we found in step 1 for \(X\) in terms of the variables \(I_n\), we can now find \(E[X]\): $$ E[X] = E\left[\sum_{n=1}^{\infty}I_n\right] = \sum_{n=1}^{\infty}E[I_n] $$ Using the result obtained in step 2 for the expected value of \(I_n\), we get: $$ E[X] = \sum_{n=1}^{\infty} P(X \geq n) = \sum_{n=0}^{\infty} P(X > n) $$ This completes the proof for part (a).

Express XY in terms of \(I_n\) and \(J_m\) sequences

For part (b), let's define a new sequence of random variables \(J_m\) for the random variable \(Y\): $$ J_{m}=\left\\{\begin{array}{ll} 1, & \text { if } m \leq Y \\\ 0, & \text { if } m>Y \end{array}\right. $$ Now, the product of \(X\) and \(Y\) can be written as the double sum of the product of \(I_n\) and \(J_m\): $$ XY = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} I_n J_m $$

Finding E[XY]

Using the expression for \(XY\) in terms of \(I_n\) and \(J_m\) found in step 4, we can find the expected value of \(XY\): $$ E[XY] = E\left[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} I_n J_m \right] = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} E[I_n J_m] $$ Now we look at the product \(I_nJ_m\) which is equal to 1 if \(n \le X\) and \(m \le Y\), and 0 otherwise. Then, $$ E[I_n J_m] = P(I_n J_m = 1) = P(X \ge n, Y \ge m) $$ Substituting this result back to the expression for \(E[XY]\) we get: $$ E[XY] = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m) $$ This completes the proof for part (b).

Key concepts

Nonnegative Integer Valued Random Variables

In probability theory, understanding the behavior of nonnegative integer valued random variables is crucial for modeling various real-world situations. These variables can only take on values that are nonnegative integers, that is, 0, 1, 2, and so on. Examples include the number of emails received in a day or the number of students attending a class.

The importance of such random variables lies in their ability to model countable events. When dealing with them, certain operations, like summation, take a specific form because of their discrete nature. As seen in the exercise, the variable's expected value, which is the long-run average outcome of a random variable, helps us understand what we can anticipate on average from the process being modeled.

A key to working with nonnegative integer valued random variables is representing them as a sum of other random variables, such as the indicator variables used in the solution, which take the value 1 or 0 based on the outcome of an event. Using indicator variables simplifies complex problems and allows for easier computation of expectations.

Probability Mass Function (PMF)

Every random variable has a distribution that shows all possible values it can take and how likely they are. For discrete random variables, like nonnegative integer valued ones, this distribution is described by a probability mass function (PMF). The PMF assigns a probability to each potential outcome of the variable, hence the term 'mass'.

The PMF is key in calculating expected values and other properties of a random variable. In the context of our exercise, the PMF helps determine the probability that a random variable exceeds a certain value, which we denote as \( P(X > n) \). In essence, it's a tool that tells us how the probabilities are 'distributed' across the different possible values of a random variable.

When working with PMFs, it’s essential to remember that the probability of all outcomes must sum to one. This fundamental property ensures that we're dealing with a valid probability distribution and sets the stage for calculating more intricate probabilities, such as those involving multiple random variables.

Convergence of Infinite Series

In probability and many other fields of mathematics, we often work with infinite series because they allow us to represent quantities that are accumulated over an indefinite number of steps. In our problem, we use infinite series to express the expected value of random variables.

An infinite series is the sum of an infinite sequence of terms. For the series to have a meaningful value, it must converge to a finite number. Convergence means that as we sum more and more terms, the total gets closer and closer to a specific value and stays there. The concept of convergence is central to ensuring the existence of an expected value because it justifies summing over an infinite number of probabilities or outcomes.

With random variables, the convergence of their associated series reassures us that we can actually compute an expected value, and it won't diverge to infinity. Thus, part of solving problems involving expected values is checking the convergence of infinite series that represent them. Through various tests and conditions, we can determine whether a series converges, which is vital for validating the results and solutions we obtain in probability theory.