Glossary

uniform – равномерный; exponential – показательный; sigma – сигма

to round off – округлить; reliability – надежность;

violation – нарушение; part – доля; temporal – временной

bell-shaped – конусообразный; to encompass – заключать

Exercises for Seminar 10

10.1. A random variable X is uniformly distributed in the interval (-2; N). Find:

(a) the differential function of the random variable X;

(b) the integral function;

(c) the probability of hit of the random variable into the interval (-1; N/2);

(d) the mathematical expectation, the dispersion and the mean square deviation the random variable X.

The answer: c) 0,5.

10.2. Buses arrive at a specified stop at 15-minute intervals starting at 7 a.m. That is, they arrive at 7, 7:15, 7:30, 7:45, and so on. If a passenger arrives at the stop at a time that is uniformly distributed between 7 and 7:30, find the probability that he waits:

(a) less than 5 minutes for a bus;

(b) more than 10 minutes for a bus.

The answer: a) 1/3; b) 1/3.

10.3. You arrive at a bus stop at 10 o’clock, 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?

The answer: a) 2/3; b) 1/3.

10.4. A random variable X is distributed under an exponential law with parameter  = 0,5. Find:

(a) the probability density and the distribution function of X;

(b) the probability of hit of the random variable X into the interval (2; 4);

(c) the mathematical expectation, the dispersion and the mean square deviation of X.

The answer: b) 0,233.

10.5. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter  = 0,5. What is

(a) the probability that a repair time exceeds 2 hours;

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

The answer: a) 0,3679; b) 0,9956.

10.6. 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 parameter 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 lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over (0, 40) (to figure – считать; to junk – утилизировать).

The answer: 0,7614; 0,75.

10.7. A normally distributed random variable X is given by the differential function: . Determine:

(a) the mathematical expectation and the dispersion of the random variable X;

(b) the probability of hit of the random variable X into the interval (3; 9).

The answer: b) 0,5328.

10.8. A random variable X is distributed under a normal law with mathematical expectation a = 20. The probability of hit of the random variable X into the interval (10; 30) is 0,6826. Determine the probability of hit of the random variable into the interval (10; 25).

The answer: 0,5328.

10.9. If X is a normal random variable with parameters a = 3 and  ² = 9, find (a) P(2 < X < 5); (b) P(X > 0); (c) P(|X – 3| > 6).

The answer: a) 0,3779; b) 0,8413; c) 0,0456.

10.10. Let X be a normal random variable with mathematical expectation 12 and dispersion 4. Find the value of C such that P(X > C) = 0,1.