Continuous random variables. Probability density

A random variable X is continuous if its function of distribution is continuous at each point and differentiable everywhere but possibly finitely many points.

The distribution function of a continuous random variable X which is differentiable everywhere but three points of break is shown at the picture.

Theorem. The probability of a separately taken value of a continuous random variable X is equal to zero, i.e. P(X = x₁) = 0 for each value x₁ of the continuous random variable X.

Corollary. If X is a continuous random variable then probability of hit of the random variable in an interval (x₁, x₂) doesn’t depend on whether the interval is open or closed, i.e.

 

The probability density (distribution density or simply density) j(x) of a continuous random variable X is the derivative of its distribution function, i.e.

 

Sometimes the probability density is said to be the differential function or the differential law of distribution.

The graph of probability density j(x) is said to be distribution curve.

Example. Let X be a random variable as in Example 2. Find the probability density of the random variable X.

Solution: The probability density i.e.

Properties of probability density

1. The probability density is a non-negative function, i.e. j(x)³ 0.

2. The probability of hit of a continuous random variable in an interval [ a, b ] is equal to the definite integral of its density in limits from a to b, i.e.

 

3. The distribution function of a continuous random variable can be expressed by the probability density:

 

4. The improper integral in infinite limits of the probability density of a continuous random variable is equal to 1:

 

Formulas of mathematical expectation and dispersion of a continuous random variable X have the following form:

(if the integral converges absolutely)

 

(if the integral converges).

 

Using D(X) = M(X²) – [M(X)]², we have D(X) = M(X²) – a² or

 

Example. Let a function j(x) be given:

Find: a) the value of the constant A for which the function is the probability density of some random variable X; b) the expression of the distribution function F(x); c) the probability that the random variable X will take on values in the segment [2; 3]; d) the mathematical expectation and the dispersion of the random variable X.

Solution: a) j(x) will be the probability density if j(x) ³ 0, i.e. A/x⁴ ³ 0 (and consequently, A ³ 0) and it satisfies the condition Consequently,

We have A = 3.

b) Find F(x): If x £ 1 then

If x > 1 then Thus,

c)

The probability P(2 £ X £ 3) can also be found as the increment of the distribution function, i.e.

d)

Since D(X) = M(X²) – a²,

Then

Glossary

distribution function – функция распределения

jump of a function – скачок функции

step discontinuous function – ступенчатая разрывная функция

increment – приращение; distribution curve – кривая распределения

probability density – плотность вероятности

 

Exercises for Seminar 8

8.1. Let the law of distribution of a discrete random variable be given:

 

X
P	0,1	0,3	0,4	0,2

 

Find the integral function of the random variable X and construct its graph.

 

8.2. Find the integral function of distribution of the random variable X – the number of hits in a target if three shots were made with the probability of hit in the target equal 0,8 for each shot.

 

8.3. A continuous random variable X is given by the integral function:

Determine:

a) the probability of hit of the random variable into the interval (2; 3);

b) the mathematical expectation, the dispersion and the mean square deviation of the random variable X.

 

8.4. A random variable X is given by the integral function:

Find the probability that in result of a trial the random variable X will take on the value:

(a) less than 0;

(b) less than 1;

(c) no less than 1;

(d) being in the interval (0; 2).

 

8.5. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given by

What is the probability that

(a) a computer will function between 50 and 150 hours before breaking down;

(b) it will function less than 100 hours?

Direction: Take e equal 2,718281.

The answer: a) 0,3834; b) 0,632.

 

8.6. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given by

What is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events E_i, i = 1, 2, 3, 4, 5, that the i ^th such tube will have to be replaced within this time, are independent.

The answer: 80/243.

 

8.7. Let X be a random variable with probability density function

(a) What is the value of C?

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

The answer: a) 3/4.

 

8.8. Compute M(X) if X has a density function given by

The answer: 4.

 

8.9. A random variable X is given by the differential function:

Find:

(a) the integral function;

(b) the probability of hit of the random variable into the interval (a /6; a /3).

The answer: b) 7/36.

 

8.10. A random variable X is given by the integral function:

Find:

(a) the differential function;

(b) the probability of hit of the random variable X into the interval (2,5; 3);

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

The answer: b) 0,599; c) M(X) = 2,566; D(X) = 0,079.

Exercises for Homework 8

8.11. Let the law of distribution of a discrete random variable be given:

 

X	– 2
p	0,4	0,3	0,2	0,1

 

Find the integral function of the random variable X and construct its graph.

 

8.12. The probability of passing the first exam by a student is 0,7, the second exam – 0,6 and the third exam – 0,8. Find the integral function of the random variable X – the number of exams passed by the student. Determine M(X).

The answer: M(X) = 2,1.

 

8.13. The density function of X is given by

If M(X) = 3/5, find a and b.

The answer: a = 3/5; b = 6/5.

 

8.14. A system consisting of one original unit and a spare can function for a random amount of time X. If the density of X is given (in units of months) by

what is the probability that the system functions for at least 5 months (a spare – запасной элемент)?

The answer: 0,616.

 

8.15. Suppose that X is a continuous random variable whose probability density function is given by

(a) What is the value of C?

(b) Find P (X > 1).

The answer: a) 3/8; b) 1/2.

 

8.16. Find M(X) and D(X) when the density function of X is

The answer: M(X) = 2/3; D(X) = 1/18.

 

8.17. Let X be a random variable with probability density function

(a) What is the value of C?

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

The answer: a) – 64/225.

 

8.18. Compute M(X) if X has a density function given by

The answer: 10.

 

8.19. A random variable X is given by the differential function:

Find:

(a) the integral function;

(b) the probability of hit of the random variable into the interval (1; 1,1).

The answer: b) 0,221.

 

8.20. A random variable X is given by the integral function:

Find:

(a) the differential function;

(b) the probability of hit of the random variable X into the interval (1; 1,5);

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

The answer: b)0,375; c) M(X) = 19/12; D(X) = 11/144.

 

 

L E C T U R E 9