Question

Let \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere, be the pmf of the random variable \(X\). Find the mgf, the mean, and the variance of \(X\).

Short answer

The moment generating function (mgf) of \(X\) is \(M(t) = \frac{1}{1 - \frac{e^t}{2}}\) for \( \frac{e^t}{2} < 1 \), its mean is 2, and its variance is also 2.

Step-by-step solution

Calculate the Moment Generating Function (mgf)

The moment generating function of a random variable \(X\) is defined as \(M(t) = E(e^{tX})\). Plug in the pmf of \(X\) into this definition and calculate the expected value. This gives \[ M(t) = \sum_{x=1}^{\infty} e^{tx} \cdot p(x) = \sum_{x=1}^{\infty} e^{tx} \cdot \left(\frac{1}{2}\right)^{x} \]

Simplify the Moment Generating Function (mgf)

Simplify the equation \(M(t) = \sum_{x=1}^{\infty} e^{tx} \cdot \left(\frac{1}{2}\right)^{x}\). Notice that \(e^{tx} \cdot \left(\frac{1}{2}\right)^{x} = \left(\frac{e^t}{2}\right)^{x}\). So this becomes a geometric series and can be simplified as follows:\[ M(t) = \frac{1}{1 - \frac{e^t}{2}}, \quad where \quad \frac{e^t}{2} < 1.\]

Calculate the Mean

The mean (or expected value) of \(X\) can be found by differentiating the mgf and evaluating it at \(t = 0\). This gives\[ \mu = \frac{d}{dt}M(t)|_{t=0} = 2. \]

Calculate the Variance

The variance of \(X\) can be found by differentiating the mgf twice and evaluating it at \(t = 0\). However, you will have to subtract \(\mu^2\) from the result since \(Var(X) = E(X^2) - [E(X)]^2 = M''(0) - \mu^2\). This gives\[ Var(X) = 2 - 2^2 = 2 \]

Key concepts

Probability Mass Function

To understand how the probability mass function (pmf) works, let's first recall that it is a function that gives the probability that a discrete random variable is exactly equal to some value. In simpler terms, it is the function that describes the distribution of probabilities across the possible values that a discrete variable can assume.

For the random variable X described in our exercise, the pmf is expressed as the function given by the formula \(p(x) = \left(\frac{1}{2}\right)^x\), but only for integer values of x starting from 1, and it is zero elsewhere. The way to interpret this pmf is that the probability of X taking on a particular value x decreases exponentially as x increases. To put it visually, imagine a plot where the height of bars represents the probability: the bars get shorter as you move to the right along the x-axis.

The fundamental property of a pmf is that the sum of all probabilities must equal 1, representing the certainty that the random variable will take on one of its possible values. This property emphasizes one of the most important aspects of the pmf, which is to show the distribution pattern of the probabilities that a discrete random variable can take.

Expected Value

The expected value, often denoted as E(X) or simply as the mean, plays a central role in probability and statistics. It signifies the long-term average value of repetitions of the experiment it represents. To find the expected value of a random variable, one must multiply each possible value of the variable by its probability and then add up all those products.

In the case of our exercise, the expected value for the random variable X with the provided pmf was determined using the moment generating function (mgf). By differentiating the mgf and evaluating at \(t = 0\), it gives us a convenient way to calculate the mean, which turned out to be 2. This means that if we were to repeat an experiment a large number of times and average the values of X we observe, we would expect to get close to 2 over the long run.

This concept of expected value is essential not only in theoretical studies but also in practical applications. It provides a summary measure that is useful when making predictions or decisions under uncertainty, as it gives a single value descriptor of a random variable's distribution.

Variance of a Random Variable

The variance is a measure of how much the values of a random variable differ from the expected value (the mean). In other words, it quantifies the dispersion or spread of the probability distribution. A smaller variance indicates that the random variable's values tend to be closer to the mean, whereas a larger variance suggests a wider spread around the mean.

In our exercise, the variance is found by first computing the second derivative of the mgf and evaluating it at \(t = 0\). This action determines the expected value of \(X^2\) (denoted as \(E(X^2)\)). However, to get the actual variance, you must subtract the square of the mean (\(\mu^2\)) from this expected value. Through these steps, the variance for X was calculated to be 2.

It is important to note that the variance is a key concept because it gives us an idea about the variability of the outcomes. In practical scenarios, knowing the variance can help in assessing the risk or uncertainty associated with a particular event or process. The variance helps to understand not only the behavior of the data but also the reliability of the mean as a measure of central tendency.