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Numerical Characteristics of Continuous Random Variables

Numerical Characteristics of Continuous Random Variables

Let a continuous random variable X be given by distribution density f(x). Take its possible values
in a closed interval [a, b].

Divide the interval [a, b] into n subintervals ∆x 1 , ∆x 2 , . . . , ∆x n and choose points x i
(i = 1, . . . , n) between a and b.
Define ME by analogy with that of the case of DRV in such a way P(x i < X < x i + ∆x i ) ≈
f(x i )∆x i :
n
∑
x i f(x i )∆x i . (6.7)
i=1
Passing to the limit, when max i ∆x i → 0 we get

n ∫ b
∑
lim x i f(x i )∆x i = xf(x)dx. (6.8)
max ∆x i →0 a
i=1

Definition 6.4. By the mean of a continuous random variable we mean the
definite integral
∫ b
E(X) = xf(x)dx. (6.9)
a ✓

If values of a continuous random variable belong to infinite interval (−∞, +∞) then a mean is
defined by the improper integral

∫
+∞
E(X) = xf(x)dx. (6.10)
−∞

Since by the variance of a random variable we mean the mean square value

E(X − E(X)) 2

of the difference X − E(X) then

∫
+∞
2
2
Var(X) = E(X − E(X)) = (X − E(x)) f(x)dx. (6.11)
−∞
If values of a continuous random variable belong to the close interval [a, b] then

b
∫
2
Var(X) = (x − E(X)) f(x)dx. (6.12)
a

Example 6.3. Given integral distribution function F(x) :


0, if x ≤ 0,

F(x) = x , if 0 < x ≤ 2,
2


1, if x > 2.
Find E(X), Var(X) and σ(X). ,

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