# Joint probability distribution of functions of random variables

• MHB
• WMDhamnekar
In summary, Leibniz's rule can be used to find the joint probability distribution of functions of random variables.
WMDhamnekar
MHB
If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation.

Hint:The joint density function can be obtained by differentiating the following equation with respect to u and v.

$P(U\leq u, V\leq v)=\iint_{(x,y):-(x+y)\leq u,\frac{x}{x+y}\leq v} f_{X,Y} (x,y) dx dy$

Now how to differentiate the above equation with respect to u and v?

If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation.

Hint:The joint density function can be obtained by differentiating the following equation with respect to u and v.

$P(U\leq u, V\leq v)=\iint_{(x,y):-(x+y)\leq u,\frac{x}{x+y}\leq v} f_{X,Y} (x,y) dx dy$

Now how to differentiate the above equation with respect to u and v?

Hello,

One can get the answer to this question below $\Downarrow$
Joint probability distribution of functions of random variables

Leibniz's rule: $$\frac{\partial}{\partial x}\int_{\alpha(x,y)}^{\beta(x,y)} f(x,y,t)dt$$
$$= f(x, y, \beta(x,y))\frac{\partial \beta}{\partial x}- f(x,y, \alpha(x,y))\frac{\partial \alpha}{\partial x}+ \int_{\alpha(x,y)}^{\beta(x,y)} \frac{\partial f(x,y,t)}{\partial x} dt$$.

Country Boy said:
Leibniz's rule: $$\frac{\partial}{\partial x}\int_{\alpha(x,y)}^{\beta(x,y)} f(x,y,t)dt$$
$$= f(x, y, \beta(x,y))\frac{\partial \beta}{\partial x}- f(x,y, \alpha(x,y))\frac{\partial \alpha}{\partial x}+ \int_{\alpha(x,y)}^{\beta(x,y)} \frac{\partial f(x,y,t)}{\partial x} dt$$.
Hello,
Would you compute the final answer by using this Leibniz's rule ? In my link, I used differential algebra and PDF of gamma random variable.

## 1. What is a joint probability distribution?

A joint probability distribution is a probability distribution that describes the likelihood of multiple random variables occurring together. It provides information about the relationship between two or more variables and how they influence each other.

## 2. What are functions of random variables?

Functions of random variables are mathematical expressions that involve one or more random variables. These functions can be used to transform the original random variables into new variables, allowing for a more comprehensive analysis of the data.

## 3. How is a joint probability distribution of functions of random variables calculated?

To calculate a joint probability distribution of functions of random variables, the functions are applied to the original random variables and the resulting values are used to create a new joint probability distribution. This can be done using mathematical formulas or statistical software.

## 4. What is the importance of joint probability distribution of functions of random variables in scientific research?

The joint probability distribution of functions of random variables is important in scientific research because it allows for a more thorough analysis of data. It can provide insights into the relationship between variables and help researchers make more accurate predictions and conclusions.

## 5. Can the joint probability distribution of functions of random variables be used for any type of data?

Yes, the joint probability distribution of functions of random variables can be used for any type of data as long as the data follows a probability distribution. It is commonly used in fields such as statistics, economics, and engineering to analyze and model complex systems and phenomena.

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