13.4 Problems on Transform Methods

Calculate directly the generating function $g_{X} \left(s\right)$ for the geometric $\left(p\right)$ distribution.

Calculate directly the generating function $g_{X} \left(s\right)$ for the Poisson $\left(\mu\right)$ distribution.

A projection bulb has life (in hours) represented by $X \sim$ exponential (1/50). The unit will be replaced immediately upon failure or at 60 hours, whichever comes first. Determine the moment generating function for the time Y to replacement.

Simple random variable X has distribution

1. Determine the moment generating function for X.
2. Show by direct calculation the $M_{X}^{'} \left(0\right) = E \left[X\right]$ and $M_{X}^{' '} \left(0\right) = E \left[X^{2}\right]$.

Use the moment generating function to obtain the variances for the following distributions

Exponential $\left(\lambda\right)$ Gamma $\left(\alpha , \lambda\right)$ Normal $\left(\mu , \sigma^{2}\right)$

The pair $\left\{X , Y\right\}$ is iid with common moment generating function $\frac{\lambda^{3}}{\left(\lambda - s\right)^{3}}$. Determine the moment generating function for $Z = 2 X - 4 Y + 3$.

The pair $\left\{X , Y\right\}$ is iid with common moment generating function $M_{X} \left(s\right) =$$\left(0 . 6 + 0 . 4 e^{s}\right)$. Determine the moment generating function for $Z = 5 X + 2 Y$.

Use the moment generating function for the symmetric triangular distribution on $\left(- c , c\right)$ as derived
in the section "Three Basic Transforms".

1. Obtain an expression for the symmetric triangular distribution on $\left(a , b\right)$ for any $a < b$.
2. Use the result of part (a) to show that the sum of two independent random variables uniform on $\left(a , b\right)$ has symmetric triangular distribution on $\left(2 a , 2 b\right)$.

Random variable X has moment generating function $\frac{p^{2}}{\left(1 - q e^{s}\right)^{2}}$.

1. Use derivatives to determine $E \left[X\right]$ and $Var \left[X\right]$.
2. Recognize the distribution from the form and compare $E \left[X\right]$ and $Var \left[X\right]$ with the result of part (a).

The pair $\left\{X , Y\right\}$ is independent. $X \sim$ Poisson (4) and $Y \sim$ geometric (0.3). Determine the generating function g_Z for $Z = 3 X + 2 Y$.

Random variable X has moment generating function

By recognizing forms and using rules of combinations, determine $E \left[X\right]$ and $Var \left[X\right]$.

Random variable X has moment generating function

By recognizing forms and using rules of combinations, determine $E \left[X\right]$ and $Var \left[X\right]$.

Suppose the pair $\left\{X , Y\right\}$ is independent, with both X and Y binomial. Use generating functions to show under what condition, if any, $X + Y$ is binomial.

Suppose the pair $\left\{X , Y\right\}$ is independent, with both X and Y Poisson.

1. Use generating functions to show under what condition $X + Y$ is Poisson.
2. What about $X - Y$? Justify your answer.

Suppose the pair $\left\{X , Y\right\}$ is independent, Y is nonnegative integer-valued, X is Poisson and $X + Y$ is Poisson. Use the generating functions to show that Y is Poisson.

If X is a nonnegative integer-valued random variable, express the generating function as a power series.

1. Show that the kth derivative at $s = 1$ is
   $g_{X}^{\left(k\right)} \left(1\right) = E \left[X \left(X - 1\right) \left(X - 2\right) \hdots \left(X - k + 1\right)\right]$
2. Use this to show the $Var \left[X\right] = g_{X}^{' '} \left(1\right) + g_{X}^{'} \left(1\right) - \left[g_{X}^{'} \left(1\right)\right]^{2}$.

Let $M_{X} \left(\cdot\right)$ be the moment generating function for X.

1. Show that $Var \left[X\right]$ is the second derivative of $e^{- s \mu} M_{X} \left(s\right)$ evaluated at $s = 0$.
2. Use this fact to show that if $X \sim N \left(\mu , \sigma^{2}\right)$, then $Var \left[X\right] = \sigma^{2}$.

Use derivatives of $M_{X_{m}} \left(s\right)$ to obtain the mean and variance of the negative binomial $\left(m , p\right)$ distribution.

Use moment generating functions to show that variances add for the sum or difference of independent random variables.

The pair $\left\{X , Y\right\}$ is iid $N \left(3 , 5\right)$. Use the moment generating function to show that $Z = 3 X - 2 Y + 3$ is is normal (see Example 3 from "Transform Methods" for general result).

Use the central limit theorem to show that for large enough sample size (usually 20 or more), the sample average

is approximately $N \left(\mu , \sigma^{2} / n\right)$ for any reasonable population distribution having mean value μ and variance σ².

A population has standard deviation approximately three. It is desired to determine the sample size n needed to ensure that with probability 0.95 the sample average will be within 0.5 of the mean value.

1. Use the Chebyshev inequality to estimate the needed sample size.
2. Use the normal approximation to estimate n (see Example 1 from "Simple Random Samples and Statistics").