6.6: Multivariable Transformations Using Jacobians

A word of caution is in order. Be sure that the bivariate transformation $u_1 = h_1(y_1, y_2), u_2 = h_2(y_1, y_2)$ is a one-to-one transformation for all $(y_1, y_2)$ such that $f_{Y_1, Y_2}(y_1, y_2) > 0$. This step is easily overlooked. If the bivariate transformation is not one-to-one and this method is blindly applied, the resulting "density" function will not have the necessary properties of a valid density function.

The multivariable transformation method is also useful if we are interested in a single function of $Y_1$ and $Y_2$ - say, $U_1 = h(Y_1, Y_2)$. Because we have only one function of $Y_1$ and $Y_2$, we can use the method of bivariate transformations to find the joint distribution of $U_1$ and another function $U_2 = h_2(Y_1, Y_2)$ and then find the desired marginal density of $U_1$ by integrating the joint density. Because we are really interested in only the distribution of $U_1$, we would typically choose the other function $U_2 = h_2(Y_1, Y_2)$ so that the bivariate transformation is easy to invert and the Jacobian is easy to work with. We illustrate this technique in the following example.

If $Y_1, Y_2, ..., Y_k$ are jointly continuous random variables and

$$U_1 = h_1(Y_1, Y_2, ..., Y_k), U_2 = h_2(Y_1, Y_2, ..., Y_k), ..., U_k = h_k(Y_1, Y_2, ..., Y_k),$$

where the transformation

$$u_1 = h_1(y_1, y_2, ..., y_k), u_2 = h_2(y_1, y_2, ..., y_k), ..., u_k = h_k(y_1, y_2, ..., y_k)$$

is a one-to-one transformation from $(y_1, y_2, ..., y_k)$ to $(u_1, u_2, ..., u_k)$ with inverse

$$y_1 = h_1^{-1}(u_1, u_2, ..., u_k), y_2 = h_2^{-1}(u_1, u_2, ..., u_k), ..., y_k = h_k^{-1}(u_1, u_2, ..., u_k),$$

and $h_1^{-1}(u_1, u_2, ...,u_k), h_2^{-1}(u_1, u_2, ...,u_k), ..., h_k^{-1}(u_1, u_2, ..., u_k)$ have continuous partial derivatives with respect to $u_1, u_2, ..., u_k$ and Jacobian

$$J = \begin{bmatrix} \frac{\partial h_1^{-1}}{\partial u_1} & \frac{\partial h_1^{-1}}{\partial u_2} & \cdots & \frac{\partial h_1^{-1}}{\partial u_k}\\ \frac{\partial h_2^{-1}}{\partial u_1} & \frac{\partial h_2^{-1}}{\partial u_2} & \cdots & \frac{\partial h_2^{-1}}{\partial u_k}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial h_k^{-1}}{\partial u_1} & \frac{\partial h_k^{-1}}{\partial u_2} & \cdots & \frac{\partial h_k^{-1}}{\partial u_k} \end{bmatrix} \neq 0,$$

then a result analogous to the one presented in this section can be used to find the joint density of $U_1, U_2, ..., U_k$. This requires the user to find the determinant of a $k \times k$ matrix, a skill that is not required in the rest of this text.

Exercises

6.63

In Example 6.14, $Y_1$ and $Y_2$ were independent exponentially distributed random variables, both with mean $\beta$. We defined $U_1 = Y_1 / (Y_1 + Y_2)$ and $U_2 = Y_1 + Y_2$ and determined the joint density of $(U_1, U_2)$ to be

$$f_{U_1, U_2}(u_1, u_2) = \begin{cases} \frac{1}{\beta^2} u_2 e^{-u_2 / \beta}, & 0 < u_1 < 1, 0 < u_2,\\ 0, & \text{otherwise.} \end{cases}$$

a) Show that $U_1$ is uniformly distributed over the interval $(0, 1)$.

b) Show that $U_2$ has a gamma density with parameters $\alpha = 2$ and $\beta$.

c) Establish that $U_1$ and $U_2$ are independent.

6.64

Refer to Exercise 6.63 and Example 6.14. Suppose that $Y_1$ has a gamma distribution with parameters $\alpha_1$ and $\beta$, that $Y_2$ is gamma distributed with parameters $\alpha_2$ and $\beta$, and that $Y_1$ and $Y_2$ are independent. Let $U_1 = Y_1 / (Y_1 + Y_2)$ and $U_2 = Y_1 + Y_2$.

a) Derive the joint density function for $U_1$ and $U_2$.

b) Show that the marginal distribution of $U_1$ is a beta distribution with parameters $\alpha_1$ and $\alpha_2$.

c) Show that the marginal distribution of $U_2$ is a gamma distribution with parameters $\alpha = \alpha_1 + \alpha_2$ and $\beta$.

d) Establish that $U_1$ and $U_2$ are independent.

6.65

Let $Z_1$ and $Z_2$ be independent standard normal random variables and $U_1 = Z_1$ and $U_2 = Z_1 + Z_2$.

a) Derive the joint density of $U_1$ and $U_2$.

b) Use Theorem 5.12 to give $E(U_1), E(U_2), V(U_1), V(U_2)$, and Cov$(U_1 , U_2)$.

c) Are $U_1$ and $U_2$ independent? Why?

d) Refer to Section 5.10. Show that $U_1$ and $U_2$ have a bivariate normal distribution. Identify all the parameters of the appropriate bivariate normal distribution.

6.66

Let $Y_1$ and $Y_2$ have joint density function $f_{Y_1, Y_2}(y_1, y_2)$ and let $U_1 = Y_1 + Y_2$ and $U_2 = Y_2$.

a) Show that the joint density of $(U_1, U_2)$ is

$$f_{U_1, U_2}(u_1, u_2) = f_{Y_1, Y_2}(u_1 - u_2, u_2).$$

b) Show that the marginal density function for $U_1$ is

$$f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1, Y_2}(u_1 - u_2, u_2) du_2.$$

c) If $Y_1$ and $Y_2$ are independent, show that the marginal density function for $U_1$ is

$$f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1}(u_1 - u_2) f_{Y_2}(u_2) du_2.$$

That is, that the density of $Y_1 + Y_2$ is the convolution of the densities $f_{Y_1}(\cdot)$ and $f_{Y_2}(\cdot)$.

6.67

Let $(Y_1, Y_2)$ have joint density function $f_{Y_1, Y_2}(y_1, y_2)$ and let $U_1 = Y_1 / Y_2$ and $U_2 = Y_2$.

a) Show that the joint density of $(U_1, U_2)$ is

$$f_{U_1, U_2}(u_1, u_2) = f_{Y_1, Y_2}(u_1 u_2, u_2) |u_2|.$$

b) Show that the marginal density function for $U_1$ is

$$f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1, Y_2}(u_1 u_2, u_2) |u_2| du_2.$$

c) If $Y_1$ and $Y_2$ are independent, show that the marginal density function for $U_1$ is

$$f_{U_1}(u_1) = \int_{-\infty}^{\infty} f_{Y_1}(u_1 u_2) f_{Y_2}(u_2) |u_2| du_2.$$

6.68

Let $Y_1$ and $Y_2$ have joint density function

$$f_{Y_1, Y_2}(y_1, y_2) = \begin{cases} 8y_1 y_2, & 0 \leq y_1 < y_2 \leq 1,\\ 0, & \text{otherwise,} \end{cases}$$

and $U_1 = Y_1 / Y_2$ and $U_2 = Y_2$.

a) Derive the joint density function for $(U_1, U_2)$.

b) Show that $U_1$ and $U_2$ are independent.

6.69

The random variables $Y_1$ and $Y_2$ are independent, both with density

$$f(y) = \begin{cases} \frac{1}{y^2}, & 1 < y,\\ 0, & \text{otherwise.} \end{cases}$$

Let $U_1 = \frac{Y_1}{Y_1 + Y_2}$ and $U_2 = Y_1 + Y_2$.

a) What is the joint density of $Y_1$ and $Y_2$?

b) Show that the joint density of $U_1$ and $U_2$ is given by

$$f_{U_1, U_2}(u_1, u_2) = \begin{cases} \frac{1}{u_1^2 (1 - u_1)^2 u_2^3}, & 1 / u_1 < u_2, 0 < u_1 < 1 / 2 \text{ and}\\ & 1 / (1 - u_1) < u_2, 1 / 2 \leq u_1 \leq 1,\\ 0, & \text{otherwise.} \end{cases}$$

c) Sketch the region where $f_{U_1, U_2}(u_1, u_2) > 0$.

d) Show that the marginal density of $U_1$ is

$$f_{U_1}(u_1) = \begin{cases} \frac{1}{2(1 - u_1)^2}, & 0 \leq u_1 < 1 / 2,\\ \frac{1}{2u_1^2}, & 1 / 2 \leq u_1 \leq 1,\\ 0, & \text{otherwise.} \end{cases}$$

e) Are $U_1$ and $U_2$ independent? Why or why not?

6.70

Suppose that $Y_1$ and $Y_2$ are independent and that both are uniformly distributed on the interval $(0, 1)$, and let $U_1 = Y_1 + Y_2$ and $U_2 = Y_1 - Y_2$.

a) Show that the joint density of $U_1$ and $U_2$ is given by

$$f_{U_1, U_2}(u_1, u_2) = \begin{cases} 1 / 2, & -u_1 < u_2 < u_1, 0 < u_1 < 1 \text{ and }\\ & u_1 - 2 < u_2 < 2 - u_1, 1 \leq u_1 < 2,\\ 0, & \text{otherwise.} \end{cases}$$

b) Sketch the region where $f_{U_1, U_2}(u_1, u_2) > 0$.

c) Show that the marginal density of $U_1$ is

$$f_{U_1}(u_1) = \begin{cases} u_1, & 0 < u_1 < 1,\\ 2 - u_1, & 1 \leq u_1 < 2,\\ 0, & \text{otherwise.} \end{cases}$$

d) Show that the marginal density of $U_2$ is

$$f_{U_2}(u_2) = \begin{cases} 1 + u_2, & -1 < u_2 < 0,\\ 1 - u_2, & 0 \leq u_1 < 1,\\ 0, & \text{otherwise.} \end{cases}$$

e) Are $U_1$ and $U_2$ independent? Why or why not?

6.71

Suppose that $Y_1$ and $Y_2$ are independent exponentially distributed random variables, both with mean $\beta$, and define $U_1 = Y_1 + Y_2$ and $U_2 = Y_1 / Y_2$.

a) Show that the joint density of $(U_1, U_2)$ is

$$f_{U_1, U_2}(u_1, u_2) = \begin{cases} \frac{1}{\beta^2} u_1 e^{-u_1 / \beta} \frac{1}{(1 + u_2)^2}, & 0 < u_1, 0 < u_2,\\ 0, & \text{otherwise.} \end{cases}$$

b) Are $U_1$ and $U_2$ independent? Why or why not?