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BBy Bot
Jun 09'24
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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 Exercise.)

BBy Bot
Jun 09'24
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An insurance company assumes that the time between claims from

each of its homeowners' policies is exponentially distributed with mean [math]\mu[/math]. It would like to estimate [math]\mu[/math] by averaging the times for a number of policies, but this is not very practical since the time between claims is about 30 years. At Galambos'[Notes 1] suggestion the company puts its customers in groups of 50 and observes the time of the first claim within each group. Show that this provides a practical way to estimate the value of [math]\mu[/math].

Notes

  1. J. Galambos, Introductory Probability Theory (New York: Marcel Dekker, 1984), p. 159.
BBy Bot
Jun 09'24
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Particles are subject to collisions that cause them to split into

two parts with each part a fraction of the parent. Suppose that this fraction is uniformly distributed between 0 and 1. Following a single particle through several splittings we obtain a fraction of the original particle [math]Z_n = X_1 \cdot X_2 \cdot\dots\cdot X_n[/math] where each [math]X_j[/math] is uniformly distributed between 0 and 1. Show that the density for the random variable [math]Z_n[/math] is

[[math]] f_n(z) = \frac 1{(n - 1)!}( -\log z)^{n - 1}. [[/math]]

Hint: Show that [math]Y_k = -\log X_k[/math] is exponentially distributed. Use this to find the density function for [math]S_n = Y_1 + Y_2 +\cdots+ Y_n[/math], and from this the cumulative distribution and density of [math]Z_n = e^{-S_n}[/math].

BBy Bot
Jun 09'24
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Assume that [math]X_1[/math] and [math]X_2[/math] are independent random variables, each

having an exponential density with parameter [math]\lambda[/math]. Show that [math]Z = X_1 - X_2[/math] has density

[[math]] f_Z(z) = (1/2)\lambda e^{-\lambda |z|}\ . [[/math]]

BBy Bot
Jun 09'24
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Suppose we want to test a coin for fairness. We flip the coin [math]n[/math]

times and record the number of times [math]X_0[/math] that the coin turns up tails and the number of times [math]X_1 = n - X_0[/math] that the coin turns up heads. Now we set

[[math]] Z= \sum_{i = 0}^1 \frac {(X_i - n/2)^2}{n/2}\ . [[/math]]

Then for a fair coin [math]Z[/math] has approximately a chi-squared distribution with [math]2 - 1 = 1[/math] degree of freedom. Verify this by computer simulation first for a fair coin ([math]p~=~1/2[/math]) and then for a biased coin ([math]p~=~1/3[/math]).

BBy Bot
Jun 09'24
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Verify your answers in Exercise Exercise(a) by computer

simulation: Choose [math]X[/math] and [math]Y[/math] from [math][-1,1][/math] with uniform density and calculate [math]Z = X + Y[/math]. Repeat this experiment 500 times, recording the outcomes in a bar graph on [math][-2,2][/math] with 40 bars. Does the density [math]f_Z[/math] calculated in Exercise \ref{exer 7.2.1}(a) describe the shape of your bar graph? Try this for Exercises Exercise(b) and Exercise Exercise(c), too.

BBy Bot
Jun 09'24
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Verify your answers to Exercise Exercise by computer

simulation.

BBy Bot
Jun 09'24
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Verify your answer to Exercise Exercise by computer

simulation.

BBy Bot
Jun 09'24
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The support of a function [math]f(x)[/math] is defined to be the set

[[math]] \{x\ :\ f(x) \gt 0\}\ . [[/math]]

Suppose that [math]X[/math] and [math]Y[/math] are two continuous random variables with density functions [math]f_X(x)[/math] and [math]f_Y(y)[/math], respectively, and suppose that the supports of these density functions are the intervals [math][a, b][/math] and [math][c, d][/math], respectively. Find the support of the density function of the random variable [math]X+Y[/math].

BBy Bot
Jun 09'24
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Let [math]X_1[/math], [math]X_2[/math], \dots, [math]X_n[/math] be a sequence of independent random

variables, all having a common density function [math]f_X[/math] with support [math][a,b][/math] (see Exercise Exercise). Let [math]S_n = X_1 + X_2 +\cdots+ X_n[/math], with density function [math]f_{S_n}[/math]. Show that the support of [math]f_{S_n}[/math] is the interval [math][na,nb][/math]. Hint: Write [math]f_{S_n} = f_{S_{n - 1}} * f_X[/math]. Now use Exercise Exercise to establish the desired result by induction.