Problem 259 - An Introduction to Mathematical Statistics and Its Applications

Question

Find the expected value of \(e^{3 X}\) if \(X\) is a binominal random variable with \(n=10\) and \(p=\frac{1}{3}\).

Step-by-Step Solution

Answer

The expected value is given by the sum \(E[e^{3X}] = \sum_{x=0}^{10} e^{3x} C(10, x)(1/3)^x(2/3)^{10-x}\).

Step 1: Identify the inputs

First, identify the inputs from the problem. \(n=10\) is the number of trials, \(p=1/3\) is the success probability, and \(g(X) = e^{3X}\) is the function of the variable.

Step 2: Calculate the expected value

Next, calculate the expected value, using the formula for expected value of a function of a random variable and the probability mass function of the binomial distribution. That is, calculate \(\sum g(x)f(x) = \sum_{x=0}^{10} e^{3x} C(10, x)(1/3)^x(2/3)^{10-x}\).

Step 3: Expand and simplify

Expand the sum by plugging in each \(x\) from 0 to 10, and simplify the expression. \(E[e^{3X}] = \sum_{x=0}^{10} e^{3x} C(10, x)(1/3)^x(2/3)^{10-x}\). This step may need to be done using a calculator or computer software.

Key Concepts

Binomial Distribution

The binomial distribution is a type of probability distribution. It applies to situations where there are fixed numbers of trials, and each trial has two possible outcomes: success or failure. It answers questions like, "What is the probability of getting exactly a certain number of successes?"
In our exercise, we use binomial distribution because we have 10 trials, each with a success probability of \( p = \frac{1}{3} \). This means each trial could result in a "success" (with probability 1/3) or "failure" (with probability 2/3).

• Number of Trials (n): The total number of trials, which is 10 in this problem.
• Success Probability (p): The likelihood of a single trial resulting in success, set at \( \frac{1}{3} \) here.

This distribution is used to model the situation and find the probabilities for different numbers of successes across trials.

Random Variables

A random variable is a variable that takes on different values based on the outcomes of a random phenomenon. It's a fundamental concept in statistics and probability theory.
In this exercise, \(X\) is our random variable, representing the number of successes in 10 trials. This number can be anywhere from 0 to 10 since all are possible outcomes.

• Discrete Random Variable: Because \(X\) can only take specific integer values (0, 1, 2,..., 10), it is known as a discrete random variable.
• Function of Random Variable: We examine \( e^{3X} \), which is a function transforming \(X\). This transformation helps us find the expected value in the context of our problem.

Understanding random variables is key because they allow us to quantify and analyze random situations.

Probability Mass Function

The probability mass function (PMF) gives the probabilities of a discrete random variable taking on each of its possible values. For a binomial distribution, PMF helps find the probability of getting exactly \( x \) successes in \( n \) trials.
In the problem, we use the PMF of the binomial distribution, expressed as \( f(x) = C(n, x) p^x (1-p)^{n-x} \). Here, \( C(n, x) \) stands for "n choose x," a combination determining how many ways \( x \) successes can occur in \( n \) trials.

• Combination Formula: \( C(n, x) = \frac{n!}{x!(n-x)!} \), where "!" denotes factorial.
• PMF in Our Problem: We calculate the overall probability for each possible outcome \(x\) from 0 to 10 using this PMF formula, allowing us to find the expected value of \( e^{3X} \).

The PMF is vital in deriving the expectations and probabilities of discrete random variables.