Problem 267 - An Introduction to Mathematical Statistics and Its Applications

Question

The form of the moment-generating function for a normal random variable is \(M_{Y}(t)=e^{a t t+b^{2} t^{2} / 2}\) (recall Example 3.12.4). Differentiate \(M_{Y}(t)\) to verify that \(a=E(Y)\) and \(b^{2}=\operatorname{Var}(Y)\)

Step-by-Step Solution

Answer

By differentiating the Moment-Generating Function of a normally distributed random variable, we can establish that \( a = E(Y) \) and \( b^2 = Var(Y) \).

Step 1: Differentiate the Moment-Generating Function

We need to calculate the first and second derivatives of the Moment-Generating Function to derive the mean (\(a\)) and variance (\(b^2\)). For a normal distribution \(Y\), the Moment-Generating Function is \(M_{Y}(t)=e^{a t t+b^{2} t^{2} / 2}\). The first derivative of \(M_{Y}(t)\) with respect to \(t\) is \(M'_{Y}(t)= e^{at + 0.5b^{2}t^{2}}(at+b^{2}t)\). Substituting \(t=0\) in \(M'_{Y}(t)\), we will get \(a=E(Y)\) since by the property of MGF, the first derivative evaluated at 0 gives the expected value.

Step 2: Calculate the second derivative

The second derivative of \(M_{Y}(t)\) with respect to \(t\) is \(M''_{Y}(t)= e^{at + 0.5b^{2}t^{2}}(at+b^{2}t)^{2} + e^{at + 0.5b^{2}t^{2}}(a+b^{2}t)\). Substituting \(t=0\) in \(M''_{Y}(t)\), we get \(a^{2}+b^{2}=Var(Y)\) since by the property of MGF, the second derivative evaluated at 0 gives the variance.

Step 3: Derive variance from second derivative

We know that variance is equal to \(E(Y^{2}) - [E(Y)]^{2}\). Since \(a=E(Y)\), substituting \(a\) in \(a^{2}+b^{2}=Var(Y)\), we get \(b^{2}=Var(Y)\). Therefore, we have successfully verified that \(b^{2}=\operatorname{Var}(Y)\).

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is a fundamental concept in statistics and is known for having specific mathematical properties.
In a normal distribution:

• The mean, median, and mode are all identical and lie at the center of the distribution.
• The distribution is symmetric about its mean.
• The spread of the curve is determined by the standard deviation, where about 68% of the data falls within one standard deviation from the mean, about 95% within two, and about 99.7% within three.

The normal distribution is also known as a Gaussian distribution and is represented by the familiar formula \[ f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
Its usefulness extends across various scientific fields due to its natural appearance in random processes, popularly known as the "bell curve."

Expected Value

The expected value is a fundamental concept in probability and statistics and refers to the mean or average of a random variable over a large number of experiments or trials. For a random variable \(Y\) with itself being normal, the expected value \(E(Y)\) represents the center of the distribution.
Using the moment-generating function (MGF), we can calculate the expected value:

• The MGF of a random variable is given by the equation \(M_Y(t) = E(e^{tY})\).
• For a normal distribution, the MGF is \(M_Y(t) = e^{\mu t + \sigma^2 t^2 / 2}\).
• By differentiating this function with respect to \(t\), and evaluating it at \(t = 0\), we find that \(E(Y) = \mu\).

This highlights how the expected value for a normal distribution is equivalent to its mean, providing a measure of central tendency for the data.

Variance

Variance measures how far a set of numbers is spread out from their average value. In the context of a normal distribution, variance quantifies the spread, or dispersion, of the data points around the mean.
For a random variable \(Y\) with a normal distribution, the variance \(Var(Y)\) is directly linked to the standard deviation \(\sigma\) by the equation: \[ Var(Y) = \sigma^2 \]The significance of variance includes the following:

• High variance indicates data points are more spread out from the mean.
• Low variance suggests data points are clustered closely around the mean.

Using the moment-generating function (MGF), we can deduce the variance by differentiating twice and evaluating it at \(t = 0\):

• The second derivative gives \(a^2 + b^2 = Var(Y)\).
• Since \(a = E(Y)\), rearranging terms will verify \(b^2 = Var(Y)\).

Variance plays a crucial role in statistics as it helps in understanding the variability and reliability of data around the mean.