Exercise 1 (No.11 on page 303)

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.)

We know the density for the minimum value of a series of independent exponential random variables with mean \(\mu\) is itself an exponential with mean \(\frac{\mu}{n}\).

So the expected time for the first of these bulbs to burn out is 10 hours, or 1000 hours divided by 100 light bulbs.


Exercise 2 (No.14 on page 303)

Assume that X1 and X2 are independent random variables, each having an exponential density with parameter λ. Show that Z = X1 − X2 has the following density \[\begin{equation} F_Z(z) = (1/2)\lambda e^{-\lambda |z|} \end{equation}\].

The probability density function for X1 and X2 exists on x greater than or equal to zero as: \[\begin{equation} F_X(x) = \lambda e^{-\lambda x} \end{equation}\]

# exponential distribution plot for X1 or X2 where lambda = 1
curve(dexp(x),xlim=c(0,15)) 

# exponential distribution plot for X1 or X2 where lambda = 1 using the raw function
curve(exp(-x),xlim=c(0,15))

# exponential distribution plot for Z = X1 - X2

#It appears this doesn't work because it's only ever looking at the same x for both functions.  We would need a different way to approximate the distribution function they are intending us to arrive at. 

curve(exp(-x)-exp(-x),xlim=c(-15,15))

# Here's the curve they're trying to have us arrive at using a lambda of 1:

curve(.5*exp(-abs(x)),xlim=c(-15,15))


Exercise 3 (No.1 on page 320)

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

  1. P(|X − 10| ≥ 2).
  2. P(|X − 10| ≥ 5).
  3. P(|X − 10| ≥ 9).
  4. P(|X − 10| ≥ 20).

Here is Chebyshev’s Inequality: \[\begin{equation} P(|X-\mu| \geq \epsilon) \leq \frac{\sigma^2}{\epsilon^2} \end{equation}\].

The probability of (a) is (100/3)/2^2, but no greater than 1, or 1 (instead of 8.333).

The probability of (b) is (100/3)/5^2, but no greater than 1, or 1 (instead of 1.333).

The probability of (c) is (100/3)/9^2, but no greater than 1, or 0.4115226.

The probability of (d) is (100/3)/20^2, but no greater than 1, or 0.8333.