Data 605 Assignment Eight

Problem 7.11

A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.) - ###10. Let X1, X2, . . . , Xn be n independent random variables each of which has an exponential density with mean µ. Let M be the minimum value of the Xj . Show that the density for M is exponential with mean µ/n. Hint: Use cumulative distribution functions

\[\mu / n = 1000/100 = 10 hours\]

Problem 7.14

Assume that X1 and X2 are independent random variables, each having an exponential density with parameter λ. Show that Z = X1 − X2 has density fZ(z) = (1/2)λe−λ|z|

\[For\ X_2 \ge X_1\] \[f_Z(z) = \int_{-\infty}^{\infty}f_{X_1}(z+x_2)f_{X_2}(x_2)dx_2 \\ f_Z(z) = \int_{-\infty}^0\lambda e^{-\lambda (z+x_2)} \lambda e^{-\lambda x_2}dx_2 \\ f_Z(z) = \int_{-\infty}^0\lambda e^{-\lambda (z)} \lambda e^{-2\lambda x_2}dx_2 \\ f_Z(z) = \lambda^2 e^{-\lambda z}(\int_{-\infty}^0 e^{-2\lambda x_2}dx_2) \\ f_Z(z) = \lambda^2 e^{-\lambda z}(\frac{ -1}{2\lambda}) \\ f_Z(z) = \frac{-\lambda}{2} (e^{-\lambda z})\]

\[For\ X_1 \ge X_2\] \[f_Z(z) = \int_{-\infty}^{\infty}f_{X_1}(z+x_2)f_{X_2}(x_2)dx_2 \\ f_Z(z) = \int_0^{\infty}\lambda e^{-\lambda (z+x_2)} \lambda e^{-\lambda x_2}dx_2 \\ f_Z(z) = \int_0^{\infty}\lambda e^{-\lambda (z)} \lambda e^{-2\lambda x_2}dx_2 \\ f_Z(z) = \lambda^2 e^{-\lambda z}(\int_0^{\infty} e^{-2\lambda x_2}dx_2) \\ f_Z(z) = \lambda^2 e^{-\lambda z}(\frac{ -1}{2\lambda}) \\ f_Z(z) = \frac{\lambda}{2} (e^{-\lambda z})\]

When we combine these two separate cases we get the combined distribution of:

\[f(z) = \frac{\lambda}{2} e^{-\lambda |z|}\]

Problem 8.1.1

1 Let X be a continuous random variable with mean µ = 10 and variance σ^2 = 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities.

\[P(|X - \mu|\ge k\sigma) \le \frac{\sigma ^2}{k^2\sigma ^2} = \frac{1}{k^2}\]

For any of the following inequalities: \[\mu=10\] \[\sigma ^2= \frac{100}{3}\] \[k=\frac{k \sigma}{\sigma}\]