Question

Find the expected value of the random variable \(\mathrm{Y}=\mathrm{f}(\mathrm{X})\), when \(\mathrm{X}\) is a discrete random variable with probability mass function \(\mathrm{g}(\mathrm{x})\). Let \(\mathrm{f}(\mathrm{X})=\mathrm{X}^{2}+\mathrm{X}+1\) and \(\operatorname{Pr}(X=x)=g(x)=\) \(\mathrm{x}=1\) \(=\quad(1 / 3) \quad x=2\) \(=\) \(\mathrm{x}=3 .\)

Short answer

The expected value of the random variable \(Y\) is given by: \(E(Y) = 1 + \frac{14}{3} + 6x\)

Step-by-step solution

In this problem, we are given the following property mass function \(g(x)\) for the random variable \(X\): \(g(1) = \frac{1}{3}\), \(g(2) = x\), and \(g(3) = =\) We are also given the function, \(f(x) = x^2 + x + 1\), that we'll use to find the expected value for the random variable \(Y = f(X)\). #Step 2: Compute probabilities for each value of \(x\)#

Before we calculate the expected value, we need to find the probabilities for each value of \(x\). Since we have \(g(x)\), we can compute this using normalization: \(g(2) = 1 - g(1) - g(3)\) \(g(2) = 1 - \frac{1}{3} - x\) #Step 3: Calculate the expected value of Y#

Now that we know the probabilities for each value of \(x\), we can calculate the expected value of \(Y\). The formula for the expected value is as follows: \(E(Y) = \sum_{x} f(x)g(x)\) \(E(Y) = f(1)g(1) + f(2)g(2) + f(3)g(3)\) #Step 4: Plug in the known values into the formula#

Next, we'll plug in the given values of \(f(x)\) and \(g(x)\) into the formula for the expected value: \(E(Y) = (1^2 + 1 + 1)(\frac{1}{3}) + (2^2 + 2 + 1)(1 - \frac{1}{3} - x) + (3^2 + 3 + 1)(x)\) #Step 5: Calculate E(Y)#

Now, we'll simplify the expression: \(E(Y) = (\frac{1}{3})(3) + (7)(\frac{2}{3} - x) + (13)(x)\) \(E(Y) = 1 + \frac{14}{3} - 7x + 13x\) Now combine like terms: \(E(Y) = 1 + \frac{14}{3} + 6x\) #Step 6: Final result#

So, the expected value of the random variable \(Y\) is given by: \(E(Y) = 1 + \frac{14}{3} + 6x\) This is the expected value of \(Y\) based on the given function and probabilities.

Key concepts

Discrete Random Variable

A discrete random variable is a type of variable that can take on a countable number of distinct values. Examples include the number of heads in a series of coin tosses, or the roll of a die. Each value the variable can assume is associated with a particular probability, and the sum total of all these probabilities is always equal to one. In the context of our exercise, the discrete random variable is denoted as X, and it can take on the values 1, 2, or 3.

To better understand, consider flipping a coin. The outcome can be either heads or tails, each with a probability of 0.5. Discrete random variables work similarly but can have more outcomes, as in the exercise's scenario, with each outcome having its own probability determined by its probability mass function.

Probability Mass Function

The probability mass function (PMF) is a function that provides the probabilities of occurrence of different possible outcomes for a discrete random variable. Essentially, the PMF, which we denote as g(x) in the exercise, maps each value x of the discrete random variable X to a probability Pr(X=x).

For instance, if we have a 6-sided die, the probability of rolling any given number between 1 and 6 is 1/6; our PMF in this case assigns a probability of 1/6 to each of the six outcomes. In our exercise, g(x) is defined for each of the possible values of X, and it should satisfy two main conditions. First, for every value of x, the probability must be non-negative; second, the sum of all probabilities for all possible values must be equal to one (this is known as the normalization condition).

Expected Value Calculation

The expected value, often denoted as E(Y), of a discrete random variable gives a measure of the central tendency of the distribution of that variable. It's akin to a long-term average—if you were to observe the value of a random variable many times and average those values, the result would approximate the expected value.

In terms of calculation, the expected value for a discrete random variable is found by multiplying each possible value the variable can take by the probability of that value occurring, then summing all those products. Mathematically, this is represented as E(Y) = \[ \sum_x f(x)g(x) \], where f(x) is a function that maps values of X to real numbers, and g(x) is the probability mass function as we've discussed. In our example, the function f(X) = X^2 + X + 1 is applied to each possible value of X, and the resulting values are then used in the expected value calculation.