Exponential law of distribution

A continuous random variable X has the exponential law of distribution with the parameter l > 0 if its probability density has the following form:

 

The exponential distribution often arises, in practice, as being the distribution of the amount of time until some specific event occurs. For instance, the amount of time (starting from now) until an earthquake occurs, or until a new war breaks out, or until a telephone call you receive turns out to be a wrong number are all random variables that tend in practice to have exponential distributions.

The distribution curve j(x) and the graph of function of distribution F(x) of the random variable X are the following:

Theorem. The function of distribution of a random variable X distributed under the exponential law is

Its mathematical expectation and its dispersion

 

From the theorem follows that for a random variable distributed under the exponential law, the mathematical expectation is equal to the mean square deviation, i.e. M(X) = s_x = 1/l.

The exponential law of distribution plays the big role in the theory of mass service and the theory of reliability.

Example. It has been established that a time of repair of TVs is a random variable X distributed under the exponential law. Determine the probability that it is required no less than 20 days on repair of a TV if the average time of repair of TVs makes 15 days. Find the probability density, the distribution function and the mean square deviation of the random variable X.

Solution: By the hypothesis the mathematical expectation M(X) = 1/l = 15, and consequently l = 1/15, and the probability density and the distribution function have the following form:

The required probability P (X ³ 20) can be found by integrating the probability density:

But it is simply to do this by using the distribution function:

Find the mean square deviation: s(X) = M(X) = 15 days.

 

Normal law of distribution

The normal law of distribution is most frequently met in practice. The main feature allocating it among other laws is that it is the limiting law to which other laws of distribution are approximated at rather frequently meeting typical conditions.

The normal distribution was introduced by the French mathematician Abraham DeMoivre in 1733 and was used by him to approximate probabilities associated with binomial random variables when the binomial parameter n is large. This result was later extended by Laplace and others and is now encompassed in a probability theory known as the central limit theorem, which is discussed later. The central limit theorem, one of the two most important results in probability theory, gives a theoretical base to the often noted empirical observation that, in practice, many random phenomena obey, at least approximately, a normal probability distribution. Some examples of this behavior are the height of a man, the velocity in any direction of a molecule in gas, and the error made in measuring a physical quantity.

A continuous random variable X has the normal law of distribution (Gauss law) with the parameters a and s ² if its probability density has the following form:

Theorem. The mathematical expectation of a random variable X distributed under the normal law is M(X) = a, and its dispersion D(X) = s ².

 

The normal law of distribution of a random variable with the parameters a = 0, s ² = 1, i.e. N (0; 1), is said to be standard or normalized.

A curve of the normal law of distribution is said to be normal or Gauss curve.

Theorem. The distribution function of a random variable X distributed under the normal law is expressed by the Laplace function F(x):

 

where .

Property 1. The probability of hit of a random variable X distributed under the normal law in the interval [ x₁; x₂ ] is

 

where

Property 2. The probability that deviation of a random variable X distributed under the normal law from the mathematical expectation a doesn’t exceed the quantity D > 0 (on absolute value) is equal to

 

where t = D/s.

The rule of three sigmas: If a random variable X has the normal distribution with the parameters a and s ², i.e. N(a; s ²), then it is practically reliable that its values are in the interval (a – 3s; a + 3s), i.e.

The violation of «the rule of three sigmas», i.e. the deviation of normally distributed random variable X is more than on 3 s (by absolute value), is an event which is practically impossible since its probability is rather small:

Example. Assuming that the height of men of a certain age group is a normally distributed random variable X with the parameters a = 173 and s ² = 36, find:

1. a) the probability density and the distribution function of the random variable X; b) parts of suits of the 4-th height (176-182) and the 3-rd height (170-176) which need to be provided in total amount of a factory for a given age group.

2. Formulate «the rule of three sigmas» for the random variable X.

Solution: 1. a) We have

b) The part of suits of the 4-th height (176 – 182 cm) in total amount of the factory is determined as probability:

since

The part of suits of the 3^rd height (170 – 176 cm) can be determined analogously, but it is simply to make by the following way:

2. It is practically reliable that the height of men of the given age group is enclosed in boundaries from a – 3s = 173 – 3 × 6 = 155 to a + 3s = 173 + 3 × 6 = 191, i.e. 155 £ X £ 191 (cm).

Remark. DeMoivre, who used the normal distribution to approximate probabilities connected with coin tossing, called it the exponential bell-shaped curve. Its usefulness, however, became truly apparent only in 1809, when the famous German mathematician K.F. Gauss used it as an integral part of his approach to predicting the location of astronomic entities. As a result, it became common after this time to call the Gaussian distribution.

During the mid to late nineteenth century, however, most statisticians started to believe that the majority of data sets would have histograms conforming to the Gaussian bell-shaped form. Indeed, it came to be accepted that it was “normal” for any well-behaved data set to follow this curve. As a result, following the lead of the British statistician Karl Pearson, people began referring to the Gaussian curve by calling it simply the normal curve.

 

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 l = 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 l = 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 s 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.

 

Exercises for Homework 10

10.11. If X is uniformly distributed over (0, 10), calculate the probability that

(a) X < 3; (b) X > 6; (c) 3 < X < 8.

The answer: a) 0,3; b) 0,4; c) 0,5.

 

10.12. 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.?

The answer: a) 2/3; b) 7/12.

 

10.13. A uniformly distributed random variable X is given by the probability density f(x) = 0,125 in the interval (a – 4; a + 4), and f(x) = 0 if otherwise. Find:

(a) the distribution function of X;

(b) the probability of hit of X into the interval (a – 3; a + 1);

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

The answer: b) 0,5.

 

10.14. A random variable X is distributed under an exponential law with parameter l = 2. Find:

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

(b) the probability of hit of the random variable X into the interval (0; 1);

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

The answer: b) 0,8647.

 

10.15. The number of years a radio functions is exponentially distributed with parameter l = 1/8. If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?

The answer: 0,3679.

 

10.16. Suppose that the length of a phone call in minutes is an exponential random variable with parameter l = 1/10. If someone arrives immediately ahead of you at a public telephone booth, find the probability that you will have to wait

(a) more than 10 minutes;

(b) between 10 and 20 minutes.

The answer: a) 0,3679; b) 0,2326.

 

10.17. A normally distributed random variable X is given by the differential function: . Find the interval in which the random variable X will hit in result of trial with the probability 0,9544.

The answer: (– 2; 2).

 

10.18. If X is a normal random variable with parameters a = 10 and s ² = 36, compute: (a) P (X > 5); (b) P (4 < X < 16); (c) P (| X – 5| > 9).

The answer: a) 0,7967; b) 0,6826; c) 0,2613.

 

10.19. Suppose that X is a normal random variable with mathematical expectation 5. If P (X > 9) = 0,2, approximately what is D(X)?

The answer: 22,5625.

 

10.20. The productivity of a winter wheat on set of allotments is distributed under a normal law with parameters: a = 50 centner/hectare, s = 10 c/h. Determine:

(a) the percentage of allotments which will have the productivity more than 40 c/h;

(b) the percentage of allotments with the productivity from 45 up to 60 c/h (productivity – урожайность; winter wheat – озимая пшеница; allotment – участок).

The answer: a) 84,13%; b) 53,28%.

 

 

L E C T U R E 11