91Ó°ÊÓ

An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of \(T\) the number of years to maturity for a randomly selected bond, is, $$ F(t)=\left\\{\begin{array}{ll} 0, & t<1 \\ \frac{1}{4}, & 1 \leq t<3, \\ \frac{1}{2}: & 3 \leq t<5, \\ \frac{3}{4}, & 5 \leq t<7 \\ 1, & t>7 \end{array}\right. $$ find (a) \(\mathrm{P}(\mathrm{T}=5)\) (b) \(P(T>3)\) (c) \(\mathrm{P}(1.4< T < 6).\)

The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function $$ F(x)=\left\\{\begin{array}{ll} 0_{+} & x<0, \\ 1-e^{-k x}, & x \geq 0. \end{array}\right. $$ Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of \(X\); (b) using the probability density function of \(X\).

A continuous random variable \(X\) that can assume values between \(x=1\) and \(x=3\) has a density function given by \(f(x)=1 / 2\). (a) Show that the area under the curve is equal to 1 . (b) Find \(\mathrm{P}(2 < X < 2.5.)\) (c) Find \(P(X \leq 1.6)\).

Find the probability distribution for the number of jazz CDs when 4 CDs are selected at random from a collection consisting of 5 jazz CDs, 2 classical CDs, and 3 rock CDs. Express your results by means of a formula.

From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement. Find the probability distribution for the total \(T\) of the 3 coins. Express the probability distribution graphically as a probability histogram.

From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls.

The time to failure in hours of an important piece of electronic equipment used in a manufactured DVD player las the density function $$ I(x)=\left\\{\begin{array}{ll} \frac{1}{2000} \exp (-a: / 2000), & x \geq 0, \\ 0, & x<0. \end{array}\right. $$ (a) Find \(F(x)\). (b) Determine the probability that the component (and thus the DVD player) lasts more than 1000 hours before the component needs to be replaced. (c) Determine the probability that the component fails before 2000 hours.

An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by $$ /(x)=\left\\{\begin{array}{ll} 3 x^{-4}, & x>1, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Verify that this is a valid density function. (b) Evaluate \(F(x)\). (c) What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers?

Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error and statisticians spend a great deal of time modeling these errors. Suppose the measurement error \(X\) of a certain physical quantity is decided by the density function $$ \| X)=\left\\{\begin{array}{ll} f c(3-x), & -1 \leq x \leq 1, \\ 0 . & \text { elsewhere. } \end{array}\right. $$ (a) Determine \(k\) that renders \(f(x)\) a valid density function. (b) Find the probability that a random error in measurement is less than \(1 / 2\). (c) For this particular measurement, it is undesirable if the magnitude of the error (i.e., \(|\mathrm{a}:|),\) exceeds 0.8 . What is the probability that this occurs?

Based on extensive testing, it is determined by the manufacturer of a washing machine that the time \(Y\) (in years) before a major repair is required is characterized by the probability density function $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{4} e^{-y / 4}, & y \geq 0, \\ 0, & \text { elsewhere. } \end{array}\right. $$ (a) Critics would certainly consider the product a bargain if it is unlikely to require a major repair before the sixth year. Comment on this by determining \(P(Y>6)\) (b) What is the probability that a major repair occurs in the first year?