12. Continuous random variables

Definition 12.1 Let be a random variable and be the function of distribution of . Then is continuous if there exists a function p (x) such that

.

Definition 12.2 The function is called the integral function of distribution and p (x) is called the differential function of distribution or density of .

Properties of

1. .

2. If a domain of is (a, b), then

a) if ;

b) increases if ;

c) if .

3. .

4. .

Properties of p (x)

1. .

2. .

.

Characteristics of continuous variables

Definition 12.3 Let be a continuous random variable with a domain . Then

a) ;

b)

c) .

13. Examples of continuous random variables.

1. Unitary law of distribution

Definition 12.4 The law of distribution of a random variable is called the unitary law if the differential function of distribution is a constant for all values x of it’s domain.

Let be a random variable with the unitary law of distribution with nonzero values at the interval . Then for all . Our problem is to find C. We have

.

So the differential function of distribution is

.

2. Exponential law of distribution

Definition 12.5 The law of distribution of a random variable is called the exponential law with a parameter if the differential function of distribution is:

Then the integral function of distribution is

.

Applying integration by parts we obtain

.

.

.

Theorem 12.1 .

3. Normal law of distribution

Definition 12.6 The law of distribution of a random variable is called the normal law with parameters if the differential function of distribution is:

.

Here we have the following formulas for characteristics of .

.

.

.

.

Now we study the function and sketch it’s graph.

a). Domain: ;

b). Range: ;

c). ;

d). , so we have

We obtain that is the point of maxima, .

e). If we use the substitution , then we obtain . As is the even function then the graph of is symmetric around the line x = a.

Theorem 12.2 Let be the normal variable with parameters . Then

.