Continuous Random Variables

Twelve percent of the population is left handed. Approximate the probability that there are at least $20$ lefthanders in a school of $200$ students. State your assumptions.

Every day Jo practices her tennis serve by continually serving until she has had a total of $50$ successful serves. If each of her serves is, independently of previous ones,

successful with probability $.4$, approximately what is the probability that she will need more than $100$ serves to accomplish her goal?

Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly $100$ times. What must be true about her first $100$ serves if she is to reach her goal?

The random variable $X$ has the probability density function

$f\left(x\right)=\left\{\begin{array}{ll}ax + bx2 & 0<x<1\\ 0& otherwise\end{array}\right.$

If $E\left[X\right] =6$, find 

(a) $P\{X<\frac{1}{2}\}$and 

(b) $Var\left(X\right)$.

Repeat Problem $6.2$ when the ball selected is replaced in the urn before the next selection.

A system consisting of one original unit plus a spare

can function for a random amount of time$X$. If the density

of $X$is given (in units of months) by

$f\left(x\right)=\left\{\begin{array}{ll}Cx{e}^{-x/2}& x>0\\ 0& x\le 0\end{array}\right.$

what is the probability that the system functions for at least $5$months? 

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by 

$f\left(x\right)=\left\{\begin{array}{ll}\frac{10}{x^2}& x>10\\ 0& x\le 10\end{array}\right.$

$\left(a\right)$Find$P\{X>20\}$

$\left(b\right)$What is the cumulative distribution function of $X?$

$\left(c\right)$) What is the probability that of$6$such types of devices, at least $3$will function for at least $15$hours? What assumptions are you making?

Let $X$ be a random variable with probability density function

$f\left(x\right)=\left\{\begin{array}{l}c(1-x^2) -1 < x < 1 \\ 0 \text{ otherwise}\end{array}\right.$

(a) What is the value of $c$?

(b) What is the cumulative distribution function of $X$?

Let be a random variable with probability density function

$f\left(x\right)=\left\{\begin{array}{ll}c\left(1-x^2\right)& -1<x<1\\ 0& otherwise\end{array}\right.$

(a) What is the value of?

(b) What is the cumulative distribution function of? 

 A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function

$f\left(x\right) = \left\{\begin{array}{ll}5{(1 - x)}^4& 0 < x < 1\\ 0& otherwise\end{array}\right.$

what must the capacity of the tank be so that the probability of the supply being exhausted in a given week is $.01?$

 Consider the function

$f\left(x\right) = \left\{\begin{array}{ll}C(2x - x^3) & 0 < x < \frac{5}{2}\\ 0 & otherwise\end{array}\right.$

Could $f$be a probability density function? If so, determine

$C$. Repeat if $f\left(x\right)$were given by

$f\left(x\right) = \left\{\begin{array}{ll} C(2x - x3)& 0 < x < \frac{5}{2}\\ 0& otherwise\end{array}\right.$

5.6. Compute$E\left[X\right]$ if $X$ has a density function given by

$\left(a\right) f\left(x\right) =\left\{\begin{array}{ll}\frac{1}{4}x{e}^{\frac{-x}{2}}& x > 0\\ 0 & otherwise\end{array}\right.$;

$\left(b\right) f\left(x\right) =\left\{\begin{array}{ll}c(1 - x^2) & -1< x < 1\\ 0& otherwise\end{array}\right.$;

$\left(c\right) f\left(x\right) =\left\{\begin{array}{ll}\frac{5}{x^2}& x > 5\\ 0& x\le 5\end{array}\right.$.

Consider Example 4b of Chapter 4, but now suppose that the seasonal demand is a continuous random variable having probability density function $f$ . Show that the optimal amount to stock is the value $s*$ that satisfies

$F\left(s^{*}\right)=\frac{b}{b+l}$

where $b$ is net profit per unit sale, $l$ is the net loss per unit

unsold, and $F$ is the cumulative distribution function of the

seasonal demand.

Trains headed for destination A arrive at the train station at $15$-minute intervals starting at 7 a.m., whereas trains headed for destination B arrive at $15$-minute intervals starting at 7:05 a.m.

(a) If a certain passenger arrives at the station at a time uniformly distributed between $7$ and $8$ a.m. and then gets on the first train that arrives, what proportion of time does he or she go to destination A?

(b)What if the passenger arrives at a time uniformly distributed

between $7:10$ and $8:10$ a.m.?

A point is chosen at random on a line segment of

length $L$. Interpret this statement, and find the probability

that the ratio of the shorter to the longer segment is

less than $\frac{1}{4}$.

A bus travels between the two cities A and B, which are $100$ miles apart. If the bus has a breakdown, the distance from the breakdown to city A has a uniform distribution over $(0, 100)$. There is a bus service station in city A, in B, and in the center of the route between A and B. It is suggested that it would be more efficient to have the three stations located $25, 50, \mathrm{and} 75$ miles, respectively, from A. Do you agree? Why?

The density function of $X$ is given by

$f\left(x\right)=\left\{\begin{array}{l}a+b{x}^2; 0\le x\le 1\\ 0; \text{Otherwise}\end{array}\right.$

If  $E\left(X\right)=\frac{3}{5}$, find $a \text{and} b$.

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads $55$ percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin $1000$ times. If the coin lands on heads $525$ or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than $525$ times, then we shall conclude that it is a fair coin.

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $.95$. Approximate the probability that at most $10$ of the next $150$ items produced are unacceptable.

Twelve percent of the population is left handed. Approximate the probability that there are at least $20$ lefthanders in a school of $200$ students. State your assumptions.

Suppose that X is a normal random variable with mean $5$. If $P\left(X>9\right)=.2$, approximately what is $Var\left(X\right)$?

In $10,000$ independent tosses of a coin, the coin landed on heads $5800$ times. Is it reasonable to assume that the coin is not fair? Explain.

Let X be a normal random variable with mean $12$and variance $4$. Find the value of $c$ such that $P\left\{X>c\right\}=0.10$ .

If $65$percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of $100$people will contain

(a) at least $50$who are in favor of the proposition;

(b) between $60$and $70$inclusive who are in favor;

(c) fewer than $75$ in favor.

You arrive at a bus stop at $10$ a.m., knowing that the bus will arrive at some time uniformly distributed between $10$ and $10:30$.

(a) What is the probability that you will have to wait longer than   $10$ minutes?

(b) If, at $10:15$, the bus has not yet arrived, what is the probability that you will have to wait at least an additional $10$ minutes?

Let $X$ be a uniform $(0,1)$ random variable. Compute $E\left[X^n\right]$ by using Proposition $2.1$, and then check the result by using the definition of expectation.

If $X$ is a normal random variable with parameters $\mu = 10$ and ${\sigma }^2 = 36$, compute

(a)  $P\left\{X>5\right\}$

(b)  $P\left\{4<X<16\right\}$

(c)  $P\left\{X<8\right\}$

(d) $P\left\{X<20\right\}$

(e) $P\left\{X>16\right\}$

The annual rainfall (in inches) in a certain region is normally distributed with $\mu = 40 \text{and} \sigma = 4$. What is the probability that starting with this year, it will take more than $10$ years before a year occurs having a rainfall of more than $50$ inches? What assumptions are you making?

Suppose that X is a normal random variable with

mean 5. If P{X > 9} = .2, approximately what is Var(X)?

The salaries of physicians in a certain specialty are approximately normally distributed. If $25$ percent of these

physicians earn less than $\backslash (180,000$ and $25$ percent earn more than $\backslash )320,000$, approximately what fraction earn

(a) less than $\backslash (200,000$?

(b) between $\backslash )280,000 \text{and} \$320,000$?

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number $6$ will appear between $150$and $200$times inclusively. If the number $6$ appears exactly $200$ times, find the probability that the number 5 will appear less than $150$ times.

Suppose that the height, in inches, of a $25$-year-old man is a normal random variable with parameters $\mu = 71 \text{and} {\sigma }^2 = 6.25$. What percentage of $25$-year-old men are taller than $6$ feet, $2$ inches? What percentage of men in the $6$-footer club are taller than $6$ feet, $5$ inches?

The lifetimes of interactive computer chips produced

by a certain semiconductor manufacturer are normally distributed with parameters$\mu = 1.4 \times {10}^6$ hours and $\sigma = 3 \times {10}^5$ hours. What is the approximate probability that a batch of $100$ chips will contain at least $20$ whose lifetimes are less than $1.8 \times {10}^6$?

A model for the movement of a stock supposes that if the present price of the stock is $s$, then after one period, it will be either $us$ with probability $p$ or $ds$ with probability $1 - p$. Assuming that successive movements are independent, approximate the probability that the stock’s price will be up at least $30$ percent after the next $1000$ periods if $u = 1.012, d = 0.990, \text{and} p = .52.$

Suppose that the life distribution of an item has the hazard rate function$\lambda \left(t\right)=t^3,t>0$. What is the probability that

 $\left(a\right)$the item survives to age$2?$ 

 $\left(b\right)$the item’s lifetime is between$.4$ and$1.4?$ 

$\left(c\right)$a $1-$year-old item will survive to age $2?$

 If$X$ is uniformly distributed over $(-1,1),$find

$\left(a\right)P\{\left|X\right|>\frac{1}{2}\}$

$\left(b\right)$the density function of the random variable$\left|X\right|$. 

If $Y$is uniformly distributed over $(0,5),$ what is the probability that the roots of the equation $4x^2+4xY+Y+2=0$are both real?

If $X$is an exponential random variable with a parameter$\lambda =1$, compute the probability density function of the random variable $Y$defined by $Y=\log \left(X\right)$

If $X$is uniformly distributed over$(0,1)$, find the density function of$Y=e^X$. 

Find the distribution of$R=A\sin \left(\theta \right)$, where $A$is a fixed constant and $\theta$is uniformly distributed on$\left(\raisebox{1ex}{$-\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$. Such a random variable $R$arises in the theory of ballistics. If a projectile is fired from the origin at an angle $\alpha$from the earth with a speed$\nu$, then the point $R$at which it returns to the earth can be expressed as$R=\left(\raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$g$}\right.\right)\sin \left(2\alpha \right)$, where $g$is the gravitational constant, equal to $980$centimeters per second squared. 

$\left(a\right)$A fire station is to be located along a road of length$A,A<\infty$. If fires occur at points uniformly chosen on$(0,A)$, where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to

minimize $E\left[\left|X-a\right|\right]$

when$X$ is uniformly distributed over $(0,A)$

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with $\mu =4$and${\sigma }^2=4$, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters$(6,9)$. A point is randomly chosen on the image and has a reading of$5$. If the fraction of the image that is black is$\alpha$, for what value of$\alpha$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region? 

The lung cancer hazard rate$\lambda \left(t\right)$ of a $t$-year-old male smoker is such that

$\lambda \left(t\right) = .027 + .00025{(t - 40)}^2$$t\ge 40$

Assuming that a $40$-year-old male smoker survives all other hazards, what is the probability that he survives to

$\left(a\right)$age$50$ and $\left(b\right)$age 60 without contracting lung cancer?

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with a parameter$\frac{1}{20}$. Smith has a used car that he claims has been driven only $10,000$miles. If Jones purchases the car, what is the

probability that she would get at least $20,000$additional miles out of it? Repeat under the assumption that the life-

time mileage of the car is not exponentially distributed, but rather is (in thousands of miles) uniformly distributed over$(0,40)$.

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameters$\lambda = \frac{1}{2}$. What is

$\left(a\right)$ the probability that a repair time exceeds $2$ hours?

$\left(b\right)$ the conditional probability that a repair takes at least $10$ hours, given that its duration exceeds$9$ hours?

The number of years a radio function is exponentially distributed with the parameter$\lambda =\frac{1}{8}$ If Jones buys a used radio, what is the probability that it will be working after an additional  $8$year?

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with $\mu = 4$ and ${\sigma }^2 = 4$, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters $(6, 9)$. A point is randomly chosen on the image and has a reading of $5$. If the fraction of the image that is black is $\alpha$, for what value of $\alpha$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region? 

(a) A fire station is to be located along a road of length $A, A<\infty$. If fires occur at points uniformly chosen on $\left(0,A\right)$, where should the station be located so as to minimize the expected distance from the fire? That is,

choose a so as to minimize $E\left[\left|X-a\right|\right]$ when X is uniformly distributed over $\left(0,A\right)$.

(b) Now suppose that the road is of infinite length— stretching from point $0$outward to $\infty$. If the distance of a fire from point $0$ is exponentially distributed with rate $\lambda$, where should the fire station now be located? That is, we want to minimize $E\left[\left|X-a\right|\right]$, where X is now exponential with rate $\lambda$.

Let $Y$ be a lognormal random variable (see Example 7e for its definition) and let $c > 0$ be a constant. Answer true or false to the following, and then give an explanation for your answer.

(a) $cY$ is lognormal;

(b) $c + Y$ is lognormal.

Use the result that for a nonnegative random variable$Y$,

$E\left[Y\right]={\int }_0^{\infty }P\left\{Y>t\right\}dt$

to show that for a nonnegative random variable$X$,

$E\left[X^n\right]={\int }_0^{\infty }n{x}^{n-1}P\left\{X>x\right\}dx$

Hint: Start with 

$E\left[X^n\right]={\int }_0^{\infty }P\left\{X^n>t\right\}dt$

and make the change of variables$t=x^n$. 

Define a collection of events$E_a,0<a<1$ having the property that $P\left(E_a\right)=1$for all$a$ but$P\left(\underset{a}{\cap }{E}_a\right)=0$

Hint: Let$X$ be uniform over$(0,1)$ and define each$E_a$ in terms of$X$.