11.5. Examples

Example 1: Discrete Distribution Let 𝑋 be a discrete random variable with: 1 1 𝑃(𝑋 = 0) = , 𝑃(𝑋= 1) = 2 2 MGF: 1 1 1 𝑀 (𝑡)= 𝐸[𝑒𝑡𝑋] = 𝑒0 + 𝑒𝑡 = (1+ 𝑒𝑡) 𝑋 2 2 2 Mean: 1 1 𝑀 (𝑡) = 𝑒𝑡 ⇒ 𝑀 (0)=
𝑋′ 2 𝑋′ 2 Variance: 1 1 1 1 2 1 𝑀 (𝑡) = 𝑒𝑡 ⇒ 𝑀 (0)= ⇒ Var(𝑋) = −( ) = 𝑋″ 2 𝑋″ 2 2 2 4

In this example, we are looking at a discrete random variable 𝑋 which can take two values: 0 or 1, with equal probabilities. The MGF (Moment Generating Function) is calculated as the expected value of the exponential function of 𝑋. Specifically, we calculate:

1. MGF: This shows how the exponential function of 𝑋 reflects the probabilities of its outcomes.

   • For 𝑃(𝑋=0) = 1/2, we have e^0 = 1.
   • For 𝑃(𝑋=1) = 1/2, we have e^t, leading to the formula for MGF: (1 + e^t)/2.

2. Mean: The mean (or expected value) of the random variable can be derived from the MGF by evaluating it at t=0. In this case, since the probabilities are equal for the outcomes, the mean results in 1/2.

3. Variance: The variance can be derived again through the MGF and correlates to how spread out 𝑋 is around its mean. After performing the necessary calculations, it can be deduced that Var(𝑋) comes out to be 1/4.

Think of throwing a very simple die that only has two faces: one that shows a '0' and another that shows '1'. Each time you throw it, you have a 50% chance of getting a '0' and a 50% chance of getting a '1'. When you calculate the MGF, it's like finding a way to sum up the 'amount of excitement' you have with each possible outcome. Averaging out these results gives you a feel for what to generally expect, which is your mean. Finally, by looking at how different your results can be from this average, you understand how 'volatile' this game is, which is represented by the variance.

Example 2: Continuous Distribution Let 𝑋 ∼ 𝑁(𝜇,𝜎2) (normal distribution) MGF: 𝜎2𝑡2 𝑀 (𝑡) = exp(𝜇𝑡+ ) 𝑋 2 Mean: 𝑀 (0)= 𝜇 𝑋′ Variance: 𝑀 (0) = 𝜇2 + 𝜎2 ⇒ Var(𝑋) = 𝜎2

In this example, we explore a continuous random variable 𝑋 that follows a normal distribution, denoted by 𝑁(𝜇,𝜎2). This means 𝑋 can take on a range of values where:

1. MGF: The moment-generating function for a normal distribution is expressed as exp(𝜇𝑡 + 1/2𝜎^2𝑡^2). It captures all moments of the distribution, providing a compact representation.

2. Mean: The expected value of the random variable 𝑋 (the mean) can be directly read from the MGF when evaluated at t=0, which yields 𝜇, indicating the central location of the distribution.

3. Variance: Similarly, variance can be derived by using the MGF's properties; in this case, the variance of a normal distribution is simply 𝜎^2, which reflects how spread out the values are around the mean (𝜇).

Imagine measuring the heights of a large group of people. If we assume their heights follow a bell-shaped curve (the normal distribution), we can be certain that most heights cluster around an average height (the mean, 𝜇) and that there’s a predictable range (or spread) of heights (the variance, 𝜎2). The MGF allows researchers to capture all the important characteristics of this height distribution in a single equation, making it easier to work with when analyzing patterns or making predictions about future height measurements.

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