MGF and Its Applications

Sure! The moment generating function (MGF) of a random variable $X$ is a function that encodes information about all of its moments. If the MGF exists in some interval around 0, then it can be used to derive the $n$-th moment of $X$.

Definition: The moment generating function $M_X(t)$ of a random variable $X$ is defined as: $M_X(t) = E[e^{tX}]$ where $E[]$ denotes the expectation (or expected value) and $t$ is a real number.

Properties:

1. The $n$-th moment of $X$, $E[X^n]$, can be found by taking the $n$-th derivative of $M_X(t)$ with respect to $t$, and then setting $t = 0$. $E[X^n] = \left. \frac{d^n}{dt^n} M_X(t) \right|_{t=0}$

2. If two random variables $X$ and $Y$ are independent, then the MGF of their sum is the product of their individual MGFs: $M_{X+Y}(t) = M_X(t)M_Y(t)$

Examples:

1. Uniform Distribution $U(a, b)$: If $X$ has a uniform distribution over the interval $[a, b]$, then the MGF of $X$ is: $M_X(t) = \frac{e^{tb} - e^{ta}}{t(b-a)}$

2. Exponential Distribution with parameter $\lambda$: If $X$ is exponentially distributed, then the MGF of $X$ is: $M_X(t) = \frac{\lambda}{\lambda - t}$ for $t < \lambda$.

3. Normal Distribution $N(\mu, \sigma^2)$: If $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, then the MGF of $X$ is: $M_X(t) = e^{\mu t + \frac{1}{2} \sigma^2 t^2}$

These MGFs can be derived by using the definition of the MGF and integrating over the respective probability densities. The MGF is a powerful tool, especially when working with sums of independent random variables or when looking for an alternative way to derive moments.

Certainly! Let's derive the moment generating function (MGF) for a normally distributed random variable $X \sim N(\mu, \sigma^2)$.

Given: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ This is the probability density function (pdf) of the normal distribution.

The MGF $M_X(t)$ is defined as: $M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx$

Now, we plug in the pdf for the normal distribution:

$M_X(t) = \int_{-\infty}^{\infty} e^{tx} \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \, dx$

To evaluate this integral, we complete the square in the exponent:

Expanding $e^{tx}$: $e^{tx} = e^{t(x-\mu+\mu)} = e^{t(x-\mu)} e^{t\mu}$

Now, we have: $M_X(t) = e^{t\mu} \int_{-\infty}^{\infty} e^{t(x-\mu)} \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \, dx$

Let's focus on the exponent by combining the exponents:

$t(x-\mu) - \frac{(x-\mu)^2}{2\sigma^2}$

Now, completing the square: $t(x-\mu) - \frac{(x-\mu)^2}{2\sigma^2} = -\frac{1}{2\sigma^2} [ (x-\mu)^2 - 2\sigma^2 tx ]$

Notice that: $(x-\mu)^2 - 2\sigma^2 tx = (x-\mu-t\sigma^2)^2 - t^2\sigma^4$

Now, plugging this into our MGF:

$M_X(t) = e^{t\mu} \int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2\sigma^2} [ (x-\mu-t\sigma^2)^2 - t^2\sigma^4 ]} \, dx$

The integral term now becomes the integral over a standard normal pdf (because of the $e^{-\frac{x^2}{2}}$ format) which is equal to 1. The second term (from completing the square) can be factored out of the integral.

Hence, we get:

$M_X(t) = e^{t\mu} e^{\frac{1}{2}t^2\sigma^2}$

Combining the exponents: $M_X(t) = e^{t\mu + \frac{1}{2}t^2\sigma^2}$

And that's the MGF for a normal random variable $X$.

Certainly! Here are the moment generating functions (MGFs) for some standard discrete distributions, including the negative binomial:

1. Bernoulli Distribution (with parameter $p$): If $X$ has a Bernoulli distribution (i.e., it can take on the values 0 or 1 with probability $p$ and $1-p$ respectively): $M_X(t) = pe^t + (1-p)$

2. Binomial Distribution (with parameters $n$ and $p$): If $X$ has a binomial distribution (i.e., $X$ is the number of successes in $n$ Bernoulli trials, each with success probability $p$): $M_X(t) = (pe^t + (1-p))^n$

3. Poisson Distribution (with parameter $\lambda$): If $X$ has a Poisson distribution (i.e., $X$ is the number of events happening in a fixed interval of time or space, with a constant mean rate of $\lambda$ events per interval): $M_X(t) = e^{\lambda(e^t-1)}$

4. Geometric Distribution (with parameter $p$): If $X$ has a geometric distribution (i.e., $X$ is the number of Bernoulli trials needed to get the first success): $M_X(t) = \frac{pe^t}{1-(1-p)e^t}$ provided that $|e^t(1-p)| < 1$.

5. Negative Binomial Distribution (with parameters $r$ and $p$): If $X$ has a negative binomial distribution (i.e., $X$ is the number of failures before the $r$th success in a series of Bernoulli trials): $M_X(t) = \left( \frac{pe^t}{1 - (1-p)e^t} \right)^r$ provided that $|e^t(1-p)| < 1$.

These moment generating functions can be derived from their respective probability mass functions by using the definition of the MGF and summing over all possible values of the random variable.

Let's assume $Y_1, Y_2, \ldots, Y_n$ are independent and identically distributed (IID) random variables with a common moment generating function (MGF) given by:

$M_{Y_1}(t)$

Now, let's define the sample mean as:

$\bar{Y} = \frac{1}{n} (Y_1 + Y_2 + \ldots + Y_n)$

To find the MGF of $\bar{Y}$, we first find the MGF of $Y_1 + Y_2 + \ldots + Y_n$, and then substitute $t/n$ for $t$ (since $M_{aX}(t) = M_X(at)$ for a constant $a$).

Given the IID property and the properties of MGFs, the MGF of the sum $S = Y_1 + Y_2 + \ldots + Y_n$ is:

$M_S(t) = M_{Y_1}(t) M_{Y_2}(t) \ldots M_{Y_n}(t) = [M_{Y_1}(t)]^n$

Now, for $\bar{Y}$:

$M_{\bar{Y}}(t) = M_S\left(\frac{t}{n}\right) = [M_{Y_1}\left(\frac{t}{n}\right)]^n$

So, that's the formula for the MGF of the sample mean of IID random variables in terms of the MGF of a single one of those variables.

The Central Limit Theorem (CLT) is a foundational result in probability and statistics. It states that, under certain conditions, the sum (or average) of a large number of independent and identically distributed (i.i.d) random variables, each with finite mean and variance, will be approximately normally distributed, regardless of the distribution of the individual variables.

Using moment generating functions (MGFs) is one approach to prove the CLT. Here's a brief outline of the proof using MGFs:

1. Setting the Scene: Let $X_1, X_2, \ldots, X_n$ be i.i.d random variables with: $E[X_i] = \mu$ $Var[X_i] = \sigma^2 < \infty$ Define: $S_n = X_1 + X_2 + \ldots + X_n$ and $Z_n = \frac{S_n - n\mu}{\sigma \sqrt{n}}$

   The aim is to show that $Z_n$ converges in distribution to a standard normal distribution as $n$ approaches infinity.

2. MGF of Standardized Variable: Find the MGF of $Z_n$: $M_{Z_n}(t) = E[e^{tZ_n}]$

3. Expanding using Independence: Utilizing the property of independence, we can express the MGF of $S_n$ in terms of the MGF of $X_i$. This can be done because the MGF of a sum of independent random variables is the product of their individual MGFs.

4. Taylor Series Expansion: The key step is to expand the MGF of $X_i$ (centered and standardized) using a Taylor series around 0, keeping the first three terms. This involves looking at the terms for the first, second, and higher moments. Due to the conditions of CLT, we know the first moment is $\mu$ and the second moment is $\sigma^2$. Higher order moments will go to zero faster as $n$ increases, under the finite variance assumption.

5. Limiting Behavior: Take the limit as $n$ approaches infinity of $M_{Z_n}(t)$. Using properties of exponential functions and the results from the Taylor series expansion, we will find that this limit is the MGF of a standard normal random variable: $\lim_{n \to \infty} M_{Z_n}(t) = e^{\frac{t^2}{2}}$

6. Concluding the Proof: Since the MGF uniquely determines the distribution of a random variable (under certain conditions, which are met here), and the limit of $M_{Z_n}(t)$ is the MGF of a standard normal distribution, we can conclude that $Z_n$ converges in distribution to a standard normal random variable as $n$ goes to infinity.

This is a rough outline, and the actual proof requires more meticulous justification at each step.

Absolutely! Let's delve deeper into Step 5.

We previously established the random variable: $Z_n = \frac{S_n - n\mu}{\sigma \sqrt{n}}$

From Step 3, its MGF is a product: $M_{Z_n}(t) = E\left[e^{\frac{t}{\sigma\sqrt{n}}(X_1 + X_2 + ... + X_n - n\mu)}\right]$

Because $X_1, X_2, \ldots, X_n$ are independent, the MGF of their sum is the product of their individual MGFs. Therefore: $M_{Z_n}(t) = \prod_{i=1}^{n} M_{\frac{X_i - \mu}{\sigma\sqrt{n}}}(t)$ $= \left(M_{\frac{X_1 - \mu}{\sigma\sqrt{n}}}(t)\right)^n$

From the Taylor series expansion around 0, the MGF of a random variable $Y$ (which has mean 0 and variance $\sigma^2$) can be approximated as: $M_Y(t) \approx 1 + tE[Y] + \frac{t^2 E[Y^2]}{2}$

Using this for our standardized $X_i$, and noting that $E[X_i - \mu] = 0$, the MGF becomes: $M_{\frac{X_i - \mu}{\sigma\sqrt{n}}}(t) \approx 1 + \frac{t^2 \sigma^2}{2n}$

Substituting this approximation into our product gives: $M_{Z_n}(t) \approx \left(1 + \frac{t^2 \sigma^2}{2n}\right)^n$

Now, as $n \to \infty$, the term inside the parentheses approaches $e^{t^2/2}$, since: $\lim_{n \to \infty} \left(1 + \frac{t^2 \sigma^2}{2n}\right)^n = e^{\frac{t^2 \sigma^2}{2}}$

But, since $\sigma^2$ is the variance of each $X_i$, we know $\sigma^2 = 1$. So, this further simplifies to: $\lim_{n \to \infty} M_{Z_n}(t) = e^{\frac{t^2}{2}}$

This is the MGF of a standard normal distribution. Thus, by the uniqueness properties of MGFs, it implies that the distribution of $Z_n$ converges to the standard normal distribution as $n$ becomes large, completing the proof.

Certainly! Step 6 serves as the culmination of the proof and leverages properties of moment generating functions (MGFs) to draw conclusions about the distribution of random variables.

Step 6: Concluding the Proof

Recap: By the end of Step 5, we found that $\lim_{n \to \infty} M_{Z_n}(t) = e^{\frac{t^2}{2}}$

This is the MGF of a standard normal random variable. The essence of Step 6 is to use the fact that if two random variables have the same MGF (over a region where the MGF exists), then those random variables have the same distribution.

Uniqueness Property of MGFs: A key property of MGFs is their uniqueness. If a random variable has an MGF that exists in an interval around 0, then that MGF uniquely determines the distribution of the random variable. In other words, if two random variables $X$ and $Y$ have MGFs $M_X(t)$ and $M_Y(t)$ respectively, and if $M_X(t) = M_Y(t)$ for all $t$ in an open interval around 0, then $X$ and $Y$ have the same distribution.

Applying the Uniqueness Property: From Step 5, we found that the limiting MGF of $Z_n$ as $n$ goes to infinity is $e^{\frac{t^2}{2}}$, which is the MGF of a standard normal random variable. Using the uniqueness property of MGFs, this means that the limiting distribution of $Z_n$ is the standard normal distribution.

Conclusion: Given that the sequence of MGFs $M_{Z_n}(t)$ converges to the MGF of a standard normal random variable, and utilizing the uniqueness of MGFs, we can infer that the distribution of $Z_n$ converges to the standard normal distribution as $n$ approaches infinity. This is the crux of the Central Limit Theorem (CLT).

Therefore, the sum (or average) of a large number of independent and identically distributed (i.i.d.) random variables, each with finite mean and variance, will be approximately normally distributed, regardless of the distribution of the individual variables.

This conclusion underpins many statistical methods and results that assume normality, especially when the sample size is large.

Alright, let's delve deeper into the gaps and nuances in the argument provided in Step 6.

1. Uniqueness Property of MGFs: As previously mentioned, MGFs, when they exist, are unique to their distributions. If two random variables have the same MGF in an interval around 0, they have the same distribution. However, not all random variables have MGFs that exist everywhere or in an interval around 0. This is a significant limitation.

2. Existence of MGFs: While the MGFs of many common distributions exist in some interval around 0, there are distributions for which the MGF does not exist in any interval around 0, or at all. For the CLT, we need a more general tool that exists for a wider class of distributions.

3. Convergence of MGFs: Even if the MGF of a sequence of random variables converges pointwise to the MGF of another random variable, this does not necessarily mean the sequence of random variables converges in distribution to the other random variable. This gap is crucial. The convergence of MGFs is a necessary but not sufficient condition for convergence in distribution.

This is where Characteristic Functions come into play. The characteristic function of a random variable $X$ is given by: $\phi_X(t) = E[e^{itX}]$ where $i$ is the imaginary unit.

Why are characteristic functions important in this context?

1. Existence: The characteristic function always exists for any random variable and for all real $t$. This is not true for MGFs.

2. Uniqueness: Like MGFs, characteristic functions are also unique. If two random variables have the same characteristic function, they have the same distribution.

3. Convergence: The Levy Continuity Theorem states that if the characteristic functions of a sequence of random variables converge pointwise to the characteristic function of another random variable, then the sequence converges in distribution to that random variable. This property does not hold for MGFs in general.

Given these advantages, the proof of the Central Limit Theorem often uses characteristic functions. Using characteristic functions, the proof is more general and avoids the pitfalls and gaps associated with MGFs.

In essence, while MGFs provide an intuitive and straightforward method to derive properties of random variables and their combinations, for results as deep and general as the CLT, we need tools that are more broadly applicable and robust, like characteristic functions.