Chapter 7, Problem 61SE

To determine

Find the joint probability density function of the sample.

Answer to Problem 61SE

The joint probability density function of the sample is,

$\underset{\_}{f\left(x_1,x_2,\mathrm{...},x_n,\lambda \right)={\displaystyle \prod _{i=1}^n\lambda e^{-\lambda x_i}}{x}_1>0,x_2>0,\mathrm{...},x_n>0}$.

Explanation of Solution

Given info:

A random sample of n transistors is taken. The transistors has a life which is exponentially distributed with parameter $\lambda$.

Calculation:

The joint probability density function of the random sample $X_1,X_2,\mathrm{...},X_n$ and the parameter $\theta$ is $f\left(x_1,x_2,\mathrm{...},x_n,\theta \right)={\displaystyle \prod _{i}f\left(x_i|\theta \right)}$.

Exponential distribution:

The random variable X is said to follow exponential distribution, if the probability density function of X is,

$f\left(x\right)=\frac{1}{\theta }{e}^{-\frac{x}{\theta }},x>0$, where $E\left(X\right)=\theta$.

The joint probability density function of exponential distribution is,

$\begin{array}{c}f\left(x_1,x_2,\mathrm{...},x_n,\lambda \right)={\displaystyle \prod _{i}f\left(x_i|\lambda \right)}\\ =\underset{\_}{{\displaystyle \prod _{i=1}^n\lambda e^{-\lambda x_i}}{x}_1>0,x_2>0,\mathrm{...},x_n>0}\end{array}$