Question

Consider \(n\) independent objects (such as light bulbs) whose failure time (i.e., lifetime) is a random variable exponentially distributed with density function \(f(x, \theta)=\theta^{-1} \exp (-x / \theta), x>0 ; 0\) for \(x \leq 0\) ( \(\theta\) is a positive parameter). 'The observations of lifetime become available in order of failure. Let $$ X_{1, n} \leq X_{2, n} \leq \cdots \leq X_{r, n} $$ dinote the lifetimes of the first \(r\) objects that fail. Determine the joint density funetion of \(X_{i, n}, i=1,2, \ldots, r\).

Short answer

The joint probability density function for the first \(r\) objects' lifetimes (the order statistics) with lifetimes \(X_{1, n}, X_{2, n}, \ldots, X_{r, n}\) that fail, where the lifetimes are exponentially distributed, is given by: $$g(x_1, x_2, \ldots, x_r; \theta) = \frac{(n-r)!}{\theta^r} \exp\left(-\frac{x_1 + x_2 + \cdots + x_r}{\theta} \right)\prod_{i=1}^r \left(1 - \exp\left(-\frac{x_i}{\theta}\right) \right).$$

Step-by-step solution

Find the Cumulative Distribution Function (CDF) of a Lifetime

Since we're given that failure time of an object is exponentially distributed, the CDF \(F(x, \theta)\) is the integral of the PDF \(f(x, \theta)\) from \(0\) to \(x\): $$F(x, \theta) = \int_0^x \theta^{-1} \exp(-t/\theta) dt$$ Solving the integral: $$F(x,\theta) = 1 - \exp(-x/\theta).$$

Compute the joint CDF of the ordered lifetimes

Now, we need to find the joint CDF of the first \(r\) failures, denoted as \(G(x_1, x_2, \ldots, x_r; \theta)\). Let \(x_1 \le x_2 \le \cdots \le x_r\) be the lifetimes of the first \(r\) objects that fail. We can find the joint CDF \(G(x_1, x_2, \ldots, x_r; \theta)\): $$G(x_1, x_2, \ldots, x_r; \theta) = \mathbb{P}(X_{1, n} \le x_1, X_{2, n} \le x_2, \ldots, X_{r, n} \le x_r)$$ To solve this, we can use the fact that the lifetimes are independent: $$G(x_1, x_2, \ldots, x_r; \theta) = \left[\mathbb{P}(X_{1, n} \le x_1)\right] \left[\mathbb{P}(X_{2, n} \le x_2 \mid X_{1, n} \le x_1)\right] \cdots \left[\mathbb{P}(X_{r, n} \le x_r \mid X_{1, n} \le x_1 ,\ldots, X_{r-1, n} \le x_{r-1})\right]$$ Now, using the CDF computed in Step 1: $$G(x_1, x_2, \ldots, x_r; \theta) = \left[1- \exp(-x_1/\theta)\right]\left[1-\exp(-x_2/\theta) - (1 - \exp(-x_1/\theta))\right]\cdots\left[1 - \exp(-x_r/\theta) - (1 - \exp(-x_{r-1}/\theta))\right]$$

Find the joint density function of the ordered lifetimes

Finally, we need to find the joint probability density function (PDF) \(g(x_1, x_2, \ldots, x_r; \theta)\), which is the partial derivative of \(G(x_1, x_2, \ldots, x_r; \theta)\) with respect to all the variables \(x_i\). The joint PDF \(g(x_1, x_2, \ldots, x_r; \theta)\) can be obtained as follows: $$g(x_1,\ldots,x_r;\theta) = \frac{\partial^r}{\partial x_1 \ldots \partial x_r} G(x_1,\ldots,x_r;\theta),$$ Computing the partial derivatives, we obtain: $$g(x_1, x_2, \ldots, x_r; \theta) = \frac{(n-r)!}{\theta^r} \exp\left(-\frac{x_1 + x_2 + \cdots + x_r}{\theta} \right)\prod_{i=1}^r \left(1 - \exp\left(-\frac{x_i}{\theta}\right) \right).$$ This is the joint probability density function for the first \(r\) objects' lifetimes (the order statistics) with lifetimes \(X_{1, n}, X_{2, n}, \ldots, X_{r, n}\) that fail, where the lifetimes are exponentially distributed.