6.3: The Method of Distribution Functions

Exercises

6.13

If $Y_1$ and $Y_2$ are independent exponential random variables, both with mean $\beta$, find the density function for their sum. (In Exercise 5.7, we considered two independent exponential random variables, both with mean $1$ and determined $P(Y_1 + Y_2 \leq 3)$.)

6.14

In a process of sintering (heating) two types of copper powder (see Exercise 5.152), the density function for $Y_1$, the volume proportion of solid copper in a sample, was given by

$$f_1(y_1) = \begin{cases} 6y_1 (1 - y_1), & 0 \leq y_1 \leq 1,\\ 0, & \text{elsewhere.} \end{cases}$$

The density function for $Y_2$, the proportion of type A crystals among the solid copper, was given as

$$f_2(y_2) = \begin{cases} 3y_2^2, & 0 \leq y_2 \leq 1,\\ 0, & \text{elsewhere.} \end{cases}$$

The variable $U = Y_1 Y_2$ gives the proportion of the sample volume due to type A crystals. If $Y_1$ and $Y_2$ are independent, find the probability density function for $U$.

6.15

Let $Y$ have a distribution function given by

$$F(y) = \begin{cases} 0, & y < 0,\\ 1 - e^{-y^2}, & y \geq 0. \end{cases}$$

Find a transformation $G(U)$ such that, if $U$ has a uniform distribution on the interval $(0, 1)$, $G(U)$ has the same distribution as $Y$.

6.16

In Exercise 4.15, we determined that

$$f(y) = \begin{cases} \frac{b}{y^2}, & y \geq b,\\ 0, & \text{elsewhere,} \end{cases}$$

is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known constant and $U$ has a uniform distribution on the interval $(0, 1)$, transform $U$ to obtain a random variable with the same distribution as $Y$.

6.17

A member of the power family of distributions has a distribution function given by

$$F(y) = \begin{cases} 0, & y < 0,\\ \left(\frac{y}{\theta} \right)^{\alpha}, & 0 \leq y \leq \theta,\\ 1, & y > \theta, \end{cases}$$

where $\alpha, \theta > 0$.

a) Find the density function.

b) For fixed values of $\alpha$ and $\theta$, find a transformation $G(U)$ so that $G(U)$ has a distribution function of $F$ when $U$ possesses a uniform $(0, 1)$ distribution.

c) Given that a random sample of size $5$ from a uniform distribution on the interval $(0, 1)$ yielded the values $.2700$, $.6901$, $.1413$, $.1523$, and $.3609$, use the transformation derived in part (b) to give values associated with a random variable with a power family distribution with $\alpha = 2, \theta = 4$.

6.18

A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution function given by

$$F(y) = \begin{cases} 0, & y < \beta,\\ 1 - \left(\frac{\beta}{y} \right)^{\alpha}, & y \geq \beta, \end{cases}$$

where $\alpha, \beta > 0$.

a) Find the density function.

b) For fixed values of $\beta$ and $\alpha$, find a transformation $G(U)$ so that $G(U)$ has a distribution function of $F$ when $U$ has a uniform distribution on the interval $(0, 1)$.

c) Given that a random sample of size $5$ from a uniform distribution on the interval $(0, 1)$ yielded the values $.0058$, $.2048$, $.7692$, $.2475$ and $.6078$, use the transformation derived in part (b) to give values associated with a random variable with a Pareto distribution with $\alpha = 2, \beta = 3.$

6.19

Refer to Exercises 6.17 and 6.18. If $Y$ possesses a Pareto distribution with parameters $\alpha$ and $\beta$, prove that $X = 1/Y$ has a power family distribution with parameters $\alpha$ and $\theta = \beta^{-1}$.

6.20

Let the random variable $Y$ possess a uniform distribution on the interval $(0, 1)$. Derive the

a) distribution of the random variable $W = Y^2$.

b) distribution of the random variable $W = \sqrt{Y}$.

6.21

Suppose that $Y$ is a random variable that takes on only integer values $1, 2, ...$. Let $F(y)$ denote the distribution function of this random variable. As discussed in Section 4.2, this distribution function is a step function, and the magnitude of the step at each integer value is the probability that $Y$ takes on that value. Let $U$ be a continuous random variable that is uniformly distributed on the interval $(0, 1)$. Define a variable $X$ such that $X = k$ if and only if $F(k - 1) < U \leq F(k), k = 1, 2, ...$. Recall that $F(0) = 0$ because $Y$ takes on only positive integer values. Show that $P(X = i) = F(i) - F(i - 1) = P(Y = i), i = 1, 2, ...$. That is, $X$ has the same distribution as $Y$. [Hint: Recall Exercise 4.5.]

6.22

Use the results derived in Exercises 4.6 and 6.21 to describe how to generate values of a geometrically distributed random variable.