Math Problems Vault

Problem 80

(a)(b)

An urn contains 7 white balls numbered $1, 2, \ldots , 7$ and 3 black balls numbered $8, 9, 10$ . Five balls are randomly selected,

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(a)

with replacement;

(b)

without replacement.

For each of the cases (a) and (b) give the distribution:

of the number of white balls in the sample;
of the minimum number in the sample;
of the maximum number in the sample;
of the minimum number of balls needed for selecting a white ball.

Problem 81

(Continuation of Problem 80.) An urn contains 7 white balls numbered $1, 2, \ldots , 7$ and 3 black balls, now numbered $1, 2$ , and $3$ (instead of $8, 9, 10$ as in Problem 80). Five balls are randomly selected, (a) with replacement, (b) without replacement. As in Problem 80, give for each of the cases (a) and (b) the distribution:

of the number of white balls in the sample;
of the minimum number in the sample;
of the maximum number in the sample;
of the minimum number of balls needed for selecting a white ball.

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Problem 82

A machine normally makes items of which $4\%$ are defective. Every hour the producer draws a sample of size 10 for inspection. If the sample contains no defective items he does not stop the machine. What is the probability that the machine will not be stopped when it has started producing items of which $10\%$ are defective.

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Problem 83

One per thousand of a population is subject to certain kinds of accident each year. Given that an insurance company has insured 5,000 persons from the population, find the probability that at most 2 persons will incur this accident.

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Problem 84

A certain airline company, having observed that $5\%$ of the persons making reservations on a flight do not show up for the flight, sells 100 seats on a plane that has 95 seats. What is the probability that there will be a seat available for every person who shows up for the flight?

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Problem 85

Workers in a factory incur accidents at the rate of two accidents per week. Calculate the probability that there will be at most two accidents,

during 1 week,
during 2 weeks,
in each of 2 weeks.

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Problem 86

Suppose that the suicide rate in a certain state is four suicides per one million inhabitants per month. Find the probability that in a certain town of population 500,000 there will be at most four suicides in a month. Would you find it surprising that during 1 year there were at least 2 months in which more than four suicides occurred?

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Problem 87

A coin is tossed 30 times. For $n = 1, 2, \ldots , 20$ , what is the conditional probability that exactly $10+n$ heads appear given that the first 10 tosses resulted in heads. Show that the conditional probability that $10+n$ tosses will result in heads, given that heads appeared at least ten times, equals

\frac{\displaystyle \binom {30}{10+n}\frac{1}{2^{n}}}{\displaystyle \sum _{k=0}^{20} \binom {30}{10+k}\frac{1}{2^{k}}}.

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Problem 88

How many children should a family have so that with probability 0.95 it has at least a boy and at least a girl.

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Problem 89

Show that the Poisson probabilities $p(k \mid \lambda )$ satisfy the recurrence relation

p(k \mid \lambda ) = \frac{\lambda }{k}p(k-1 \mid \lambda ),

and hence determine the values of $k$ for which the terms $p(k \mid \lambda )$ reach their maximum (for given $\lambda$ ).

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Problem 90

(a)(b)(c)(d)

Verify that each of the following functions $f$ is a probability density function and sketch its graph.

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(a)

$\displaystyle f(x) = 1-\left|1-x\right| \qquad \text{for } 0 < x < 2$

(b)

$\displaystyle f(x) = \frac{1}{\pi }\frac{\beta }{\beta^{2}+(x-a)^{2}} \qquad \text{for } -\infty < x < \infty ,$

(c)

$\displaystyle f(x) = \frac{1}{2\sigma }e^{-(\left|x-\mu \right|)/\sigma } \qquad \text{for } -\infty < x < \infty ,$

(d)

$\displaystyle f(x) = \tfrac {1}{4}xe^{-x/2} \qquad \text{for } 0 < x < \infty .$

Problem 91

The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variable with density

f(x) = \begin{cases} cx & \text{for } 0 \leq x < 3, \\ c(6-x) & \text{for } 3 \leq x < 6, \\ 0 & \text{otherwise.} \end{cases}

Find the value of $c$ which makes $f$ a probability density function.
What is the probability that the number of kilos of bread that will be sold in a day is, (a) more than 300 kilos? (b) between 150 and 450 kilos?
Denote by $A$ and $B$ the events in (a) and (b), respectively. Are $A$ and $B$ independent events?

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Problem 92

(a)(b)(c)(d)

Suppose that the duration in minutes of long-distance telephone conversations follows an exponential density function;

f(x) = \tfrac {1}{5}e^{-x/5} \qquad \text{for } x > 0.

Find the probability that the duration of a conversation:

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(a)

will exceed 5 minutes;

(b)

will be between 5 and 6 minutes;

(c)

will be less than 3 minutes;

(d)

will be less than 6 minutes given that it was greater than 3 minutes.

Problem 93

(a)(b)(c)

A number is randomly chosen from the interval $(0, 1)$ . What is the probability that:

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(a)

its first decimal digit will be a 1;

(b)

its second decimal digit will be a 5;

(c)

the first decimal digit of its square root will be a 3?

Problem 94

The height of men is normally distributed with mean $\mu = 167$ cm and standard deviation $\sigma = 3$ cm.

What is the percentage of the population of men that have height,
1. greater than 167 cm,
2. greater than 170 cm,
3. between 161 cm and 173 cm?
In a random sample of four men what is the probability that:
1. all will have height greater than 170 cm;
2. two will have height smaller than the mean (and two bigger than the mean)?

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Problem 95

(a)(b)

A machine produces bolts the length of which (in centimeters) obeys a normal probability law with mean 5 and standard deviation $\sigma = 0.2$ . A bolt is called defective if its length falls outside the interval $(4.8, 5.2)$ .

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(a)

What is the proportion of defective bolts that this machine produces?

(b)

What is the probability that among ten bolts none will be defective?

Problem 96

(a)(b)(c)

Consider a shop at which customers arrive at random at a rate of twenty persons per hour. What is the probability that the time intervals between succesive arrivals will be:

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(a)

shorter than 3 minutes;

(b)

longer than 4 minutes.

(c)

Suppose that $10\%$ of the customers buy a certain object. Find the distribution of the number of customers who buy an object in an hour.

Problem 97

If $X$ is a continuous random variable with cumulative distribution function $F$ and density function $f$ , show that the random variable $Y = X^{2}$ is also continuous and express its cumulative distribution function and density in terms of $F$ and $f$ .

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Problem 98

(a)(b)(c)

(Continuation of Problem 97.) Find the density of $Y = X^{2}$ when $X$ has:

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(a)

the normal distribution $N(\mu , \sigma^{2})$ ;

(b)

the Laplace distribution (see (2.9));

(c)

the Cauchy distribution (see (2.12)).

Problem 99

(a)(b)(c)

As in Exercises 97 and 98 with $Y = \left|X\right|$ .

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(a)

(b)

(c)

Problem 100

If the $\log X$ is normally distributed then $X$ is said to have a lognormal distribution. Find its density.

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