Suppose now that X is a gamma random variable, which is continuous, and
has density function
f (x) = ba
Γ(a) xa−1 e−bx, 0 < x < ∞,
where Γ(a) = ∫ ∞
0 xa−1 e−x dx. Here, a > 0 and b > 0.
Find
GX (t) = E(etX )
and for what values of t are appropriate.
How could you use GX to find E(X) and Var(X).

We are given that \( X \) is a gamma-distributed random variable with the probability density function (PDF):

\[
f(x) = \frac{b^a}{\Gamma(a)} x^{a-1} e^{-bx}, \quad 0 < x < \infty,
\]
where \( a > 0 \) and \( b > 0 \), and \( \Gamma(a) \) is the Gamma function:

\[
\Gamma(a) = \int_0^\infty x^{a-1} e^{-x} dx.
\]

We want to compute the moment-generating function (MGF) or Laplace transform \( G_X(t) = E(e^{tX}) \) and use it to find the mean \( E(X) \) and variance \( \text{Var}(X) \).

### 1. Moment-Generating Function (MGF)

The moment-generating function \( G_X(t) \) is defined as the expected value of \( e^{tX} \):

\[
G_X(t) = E(e^{tX}) = \int_0^\infty e^{tx} f(x) dx.
\]

Substitute the PDF of the gamma distribution:

\[
G_X(t) = \int_0^\infty e^{tx} \frac{b^a}{\Gamma(a)} x^{a-1} e^{-bx} dx.
\]

This simplifies to:

\[
G_X(t) = \frac{b^a}{\Gamma(a)} \int_0^\infty x^{a-1} e^{-(b-t)x} dx.
\]

The integral \( \int_0^\infty x^{a-1} e^{-cx} dx \) is known as the Gamma function \( \Gamma(a) \), evaluated at \( c \):

\[
\int_0^\infty x^{a-1} e^{-cx} dx = \frac{\Gamma(a)}{c^a}, \quad \text{for } c > 0.
\]

Here, \( c = b - t \), so we require \( b - t > 0 \) or \( t < b \). Applying this result:

\[
G_X(t) = \frac{b^a}{\Gamma(a)} \cdot \frac{\Gamma(a)}{(b-t)^a}.
\]

Simplifying:

\[
G_X(t) = \left( \frac{b}{b-t} \right)^a, \quad \text{for } t < b.
\]

Thus, the moment-generating function is:

\[
G_X(t) = \left( \frac{b}{b-t} \right)^a, \quad t < b.
\]

### 2. Finding \( E(X) \) and \( \text{Var}(X) \)

To find the mean \( E(X) \) and variance \( \text{Var}(X) \), we use the properties of the MGF:

- The mean \( E(X) \) is given by:

\[
E(X) = G_X'(0) = \left. \frac{d}{dt} \left( \frac{b}{b-t} \right)^a \right|_{t=0}.
\]

First, differentiate \( G_X(t) = \left( \frac{b}{b-t} \right)^a \) with respect to \( t \):

\[
G_X'(t) = a \left( \frac{b}{b-t} \right)^{a+1} \cdot \frac{1}{(b-t)^2}.
\]

At \( t = 0 \):

\[
G_X'(0) = a \cdot \frac{1}{b}.
\]

Thus, the mean is:

\[
E(X) = \frac{a}{b}.
\]

- The second moment \( E(X^2) \) is found by differentiating the MGF twice:

\[
E(X^2) = G_X''(0).
\]

The variance is:

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

Differentiating the MGF a second time:

\[
G_X''(t) = a(a+1) \left( \frac{b}{b-t} \right)^{a+2} \cdot \frac{1}{(b-t)^4}.
\]

At \( t = 0 \):

\[
G_X''(0) = a(a+1) \cdot \frac{1}{b^2}.
\]

Thus:

\[
E(X^2) = \frac{a(a+1)}{b^2}.
\]

Finally, the variance is:

\[
\text{Var}(X) = E(X^2) - \left( E(X) \right)^2 = \frac{a(a+1)}{b^2} - \left( \frac{a}{b} \right)^2 = \frac{a}{b^2}.
\]

### Conclusion:

- The mean \( E(X) = \frac{a}{b} \).
- The variance \( \text{Var}(X) = \frac{a}{b^2} \).

### Do you want any further details or have any questions?

Here are 5 related questions:
1. How would the MGF change if \( a \) or \( b \) were negative?
2. How can the MGF be used to calculate higher moments (e.g., skewness)?
3. What are the conditions on the parameters \( a \) and \( b \) to ensure the existence of the MGF?
4. How does the Gamma distribution relate to other distributions, such as the exponential distribution?
5. Can you derive the MGF for other well-known distributions, such as the normal distribution?

**Tip:** The moment-generating function can be used to derive all moments of a random variable by differentiating it successively at \( t = 0 \).

Suppose now that X is a gamma random variable, which is continuous, and has density function f(x) = (b^a / Γ(a)) * x^(a−1) * e^(−bx), 0 < x < ∞, where Γ(a) = ∫₀^∞ x^(a−1) * e^(−x) dx. Here, a > 0 and b > 0. Find G_X(t) = E(e^(tX)) and for what values of t are appropriate. How could you use G_X to find E(X) and Var(X)?

Explore how to derive the moment-generating function (MGF) for a gamma-distributed random variable. Learn how the MGF helps calculate the mean (E(X) = a / b) and variance (Var(X) = a / b²). Suitable for undergraduate students studying probability and statistics.

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