Lesson 1.11: Masterclass on Continuous MGFs

This lesson completes our single-variable toolkit by applying the Moment Generating Function (MGF) to the continuous domain. We provide full analytical derivations for the MGFs of the Uniform, Exponential, and the all-important Normal distribution. We will then use these 'fingerprints' to generate their key moments, laying the rigorous mathematical bedrock for the Central Limit Theorem and advanced risk modeling.

Part 1: Upgrading the MGF for the Continuous World

The Core Principle Remains the Same

The definition of the MGF is universal: $M_X(t) = E[e^{tX}]$ . The only thing that changes is the tool we use to calculate the expectation. We swap summation for integration.

Definition: MGF for Continuous R.V.s

For a continuous random variable $X$ with PDF $f(x)$ , the MGF is:

M_X(t) = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx

The Moment Generation Rule

The magic of the MGF is that once the (often difficult) integration is done, finding moments becomes a simple act of differentiation. The core property is unchanged:

E[X^k] = M^{(k)}_X(0) = \left. \frac{d^k}{dt^k} M_X(t) \right|_{t=0}

Part 2: Rigorous Derivations of Foundational MGFs

1. The Uniform(

a, b

) Distribution (The Warm-up)

PDF is

f(x) = 1/(b-a)

for

x \in [a, b]

Uniform MGF Derivation

M_X(t) = \int_a^b e^{tx} \frac{1}{b-a} dx = \frac{1}{b-a} \left[ \frac{e^{tx}}{t} \right]_a^b

M_X(t) = \frac{e^{tb} - e^{ta}}{t(b-a)}

Uniform Moment Derivations

Finding derivatives requires L'Hopital's rule. It's complex, so we often state the moments which were derived more easily in Lesson 1.9.

Mean: $E[X] = (a+b)/2$

Variance: $\text{Var}(X) = (b-a)^2/12$

Skewness: $0$ (The distribution is perfectly symmetric).

Excess Kurtosis: $-6/5$ (It has "thinner" tails than a Normal distribution).

2. The Exponential(

\lambda

) Distribution (The Workhorse)

PDF is

f(x) = \lambda e^{-\lambda x}

for

x \ge 0

Exponential MGF Derivation

M_X(t) = \int_0^\infty e^{tx} (\lambda e^{-\lambda x}) dx = \lambda \int_0^\infty e^{(t-\lambda)x} dx

= \lambda \left[ \frac{e^{(t-\lambda)x}}{t-\lambda} \right]_0^\infty

For this integral to converge, the exponent must be negative, so we require $t-\lambda < 0 \implies t < \lambda$ .

= \frac{\lambda}{t-\lambda} (0 - e^0) = \frac{\lambda}{t-\lambda}(-1)

M_X(t) = \frac{\lambda}{\lambda - t}

Exponential Moment Derivations

Let's find the first two moments by differentiation. We'll write $M(t) = \lambda(\lambda-t)^{-1}$ .

Mean:

M'_X(t) = (-1)\lambda(\lambda-t)^{-2}(-1) = \lambda(\lambda-t)^{-2}

E[X] = M'_X(0) = \lambda(\lambda)^{-2} = 1/\lambda

Variance:

M''_X(t) = (-2)\lambda(\lambda-t)^{-3}(-1) = 2\lambda(\lambda-t)^{-3}

E[X^2] = M''_X(0) = 2\lambda(\lambda)^{-3} = 2/\lambda^2

\text{Var}(X) = E[X^2] - (E[X])^2 = \frac{2}{\lambda^2} - \left(\frac{1}{\lambda}\right)^2 = \frac{1}{\lambda^2}

3. The Normal(

\mu, \sigma^2

) Distribution (The Main Event)

The most important derivation for a Quant or ML professional.

Normal MGF Derivation (No-Skip Masterclass)

We start with the MGF of a standard normal $Z \sim N(0,1)$ . Its PDF is $\phi(z) = \frac{1}{\sqrt{2\pi}}e^{-z^2/2}$ .

M_Z(t) = E[e^{tZ}] = \int_{-\infty}^\infty e^{tz} \frac{1}{sqrt{2\pi}}e^{-z^2/2} dz

Step 1: Combine the exponents.

= \frac{1}{sqrt{2\pi}} \int_{-\infty}^\infty \exp\left(-\frac{1}{2}(z^2 - 2tz)\right) dz

Step 2: The "Completing the Square" Trick. We want to make the exponent look like $-(z-t)^2/2 = -(z^2 - 2tz + t^2)/2$ . To do this, we add and subtract $t^2$ inside the parenthesis.

= \frac{1}{sqrt{2\pi}} \int_{-\infty}^\infty \exp\left(-\frac{1}{2}(z^2 - 2tz + t^2 - t^2)\right) dz

= \frac{1}{sqrt{2\pi}} \int_{-\infty}^\infty \exp\left(-\frac{(z-t)^2}{2} + \frac{t^2}{2}\right) dz

Step 3: Separate the exponents and factor out the constant. The $e^{t^2/2}$ term is a constant with respect to $z$ .

= e^{t^2/2} \int_{-\infty}^\infty \frac{1}{sqrt{2\pi}} \exp\left(-\frac{(z-t)^2}{2}\right) dz

Step 4: Recognize the remaining integral. The entire integral that remains is the PDF of a Normal distribution with mean $t$ and variance 1. The total area under any PDF is exactly 1.

M_Z(t) = e^{t^2/2} cdot (1) = e^{t^2/2}

Step 5: Generalize to $X \sim N(\mu, \sigma^2)$ . We use the property $X = \mu + \sigma Z$ and $M_{aX+b}(t) = e^{tb}M_X(at)$ .

M_X(t) = M_{mu + sigma Z}(t) = e^{tmu} M_Z(sigma t) = e^{tmu} e^{(sigma t)^2/2}

M_X(t) = e^{mu t + sigma^2 t^2 / 2}

Normal Moment Derivations

Let $M(t) = e^{\mu t + \sigma^2 t^2 / 2}$ .

Mean:

M'(t) = M(t) cdot (mu + sigma^2 t)

E[X] = M'(0) = e^0(mu + 0) = mu

Variance:

M''(t) = M'(t)(mu + sigma^2 t) + M(t)(sigma^2)

E[X^2] = M''(0) = (mu)(mu) + (1)(sigma^2) = mu^2 + sigma^2

\text{Var}(X) = (mu^2 + sigma^2) - (mu)^2 = sigma^2

Skewness: The distribution is perfectly symmetric, so its standardized skewness is **0**.

Kurtosis: The fourth central moment is $E[(X-\mu)^4] = 3\sigma^4$ . This gives a kurtosis of 3, and an **excess kurtosis of 0**. This is the benchmark against which all other distributions' "tail-heaviness" is measured.

A Note on 'Fat-Tailed' Distributions

You might wonder about distributions like the Student's t or Lognormal. A fascinating and critical feature of these distributions is that their MGFs **do not exist** in the traditional sense, because the defining integral $E[e^{tX}]$ diverges (goes to infinity). This mathematical fact is the very reason they are useful for modeling extreme financial risk—their "moments" can be infinite, which is the definition of a "fat tail." We will explore these in Module 2.

What's Next? From One to Many

We have reached the summit of single-variable probability theory. We have a complete toolbox of discrete and continuous distributions and a master tool (the MGF) for analyzing them with calculus.

But the real world is not about one variable; it's about how multiple variables interact. How does one stock's return relate to another's? This is the domain of multi-variable probability, and it's the gateway to understanding correlation, covariance, and regression.

Lesson 1.10: The Continuous Toolbox

Lesson 1.12: Thinking in Multiple Dimensions: Joint Distributions