Page 38 - 4660

Discrete random variables

Numerical Characteristics of DRV

In what follows we consider such numerical characteristics:

• Mathematical expectation or mean value.

• Dispersion or variance.

• Standard deviation.

Mean and its Properties

Definition 5.9. By the mathematical expectation (or mean) of a DRV X we
mean the sum of products of all its values and corresponding probabilities, that
is
∑
E(X) = x i p i . (5.6)
✓
i
The alternative notations M(X) and ⟨x⟩ and µ are also commonly used to denote the mean. If
in (5.6) the series is not absolutely convergent, we say that the distribution does not have a mean,
but this is very rare in technical applications.
Let us compute a mean of the DRV and the average win for one ticket in the Example 5.1

10 1 89
E(X) = 1 · + 50 · + 0 · = 0.6.
100 100 100
Average win is 10+50 = 0.6.
100
So we can see that the probability meaning of a mathematical expectation is the mean value
of a random variable.
Using the formula (5.6) it is possible to prove the following properties of mean (ME).

1. EC = C,

2. C(CX) = CE(X) for an arbitrary constant C;

3. E(X ± Y ) = E(X) ± E(Y ) for arbitrary random variables X and Y ;

4. E(XY ) = E(X) · E(Y ) for independent random variables X and Y,

5. E(X − E(X)) = 0.

Variance and its Properties

Definition 5.10. By the variance (or dispersion) of a random variable X
2
denoted by Var(x) (or D(X)) we mean the mean square value E(X − E(x)) of the
difference X − E(x). So

∑
2
2
Var(X) = E(X − E(X)) = (x j − µ) p j . (5.7)
j
where µ has been written for the expectation value E(X) of X. ✓

2
The alternative notations are V [x] (variance) and σ . As in the case of the mean, unless the
series in (5.7) converge the distribution does not have a variance.

38