A random variable X has the gamma distribution f(x)=(λ)/(Γ(r))(λx)^r-1 e^-λx, x>0 (a) Show tha

step 1 For exercise 99 part a, we have to find the moment generating function. So we have mx, m sub x o

A random variable X has the gamma distribution...

A random variable $X$ has the gamma distribution
$$
f(x)=\frac{\lambda}{\Gamma(r)}(\lambda x)^{r-1} e^{-\lambda x}, \quad x>0
$$
(a) Show that the moment-generating function of $X$ is
$$
M_{X}(t)=\left(1-\frac{t}{\lambda}\right)^{-r}
$$
(b) Find the mean and variance of $X$.

Key Concepts

Gamma Distribution

The gamma distribution is a two-parameter family of continuous probability distributions characterized by a shape parameter and a rate (or scale) parameter. It is widely used in statistics to model waiting times and is noted for its flexibility in modeling skewed distributions. Its probability density function is defined using the gamma function, which allows the distribution to generalize other distributions like the exponential and chi-square distributions.

Moment Generating Function (MGF)

The moment generating function of a random variable is a function that, when it exists, uniquely characterizes the distribution of the variable. It is defined as the expected value of the exponential function of a scaling of the random variable. The MGF is particularly useful because its derivatives at zero yield the moments of the distribution, providing a systematic way to compute the mean, variance, and higher-order moments.

Gamma Function

The gamma function extends the notion of factorials to the complex plane (excluding non-positive integers). For any positive number, it is defined via an integral representation and is used in the formulation of the gamma distribution. It is a key component in the normalization factor of the gamma distribution's density function, ensuring that the total probability integrates to one.

Calculation of Moments

The moments of a random variable, such as the mean and variance, can be calculated by taking the derivatives of its moment generating function. Specifically, the first derivative of the MGF at zero gives the mean, while the second derivative, along with the first, is used to compute the variance. This process of differentiation leverages the power series expansion of the MGF to extract moment information efficiently.

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