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Numerical Characteristics of DRV

Theorem 5.1. The variance also can be found by the formula

2
2
Var(X) = E(X ) − E (X).
PROOF.

2
2
2
Var(X) = E(X − E(X)) = E(X − 2E(X) · X + E (X)) =
2
2
2
2
= E(X ) − 2E(X) · E(X) + E(E(X )) = E(X ) − E (X). 2
The following properties of a variance are valid.
1. A variance of a constant is equal to zero: Var(C) = 0.
2. Var(X) ≥ 0.

2
3. Var(CX) = C Var(X) for an arbitrary constant C.
4. Var(X ± Y ) = Var(X) + Var(Y ) for independent random variables X and Y.

2
2
2
2
5. Var(XY ) = E(X )E(Y ) − E (X)E (Y ) for independent random variables X and Y.
Standard deviation
Definition 5.11. By mean square deviation or standard deviation we mean

√
σ(X) = Var(X). (5.8)
✓

Roughly speaking, σ measures the spread (about x = µ) of the values that X can assume.

Example 5.3. Find the standard deviation of the data in the following table
X −2 −1 0 1 2
P 0.1 0.1 0.5 0.1 0.2 ,

2
Solution. For DRV X we get
X 2 4 1 0 1 4
P 0.1 0.1 0.5 0.1 0.2
2
2
2
Hence E(X) = 0.2, E(X ) = 1.4 and Var(X) = E(X ) − E (X) = 1.4 − 0.04 = 1.36, so
√ √
σ(X) = Var(X) = 1.36.
Mode and median. Although the mean discussed in the above paragraphs is the most common
measure of the ’average’ of a distribution, two other measures, which do not rely on the concept
of expectation values, are frequently encountered.
The mode of a distribution is the value of the random variable X at which the probability
(density) function f(x) has its greatest value. If there is more than one value of X for which this is
true then each value may equally be called the mode of the distribution.
The median M of a distribution is the value of the random variable X at which the cumulative
1
1
probability function F(x) takes the value , i.e. F(M) = . Related to the median are the lower
2 2
3
1
and upper quartiles Q 1 and Q 3 of the PDF, which are defined such that F(Q 1 ) = , F(Q 3 ) = .
4 4
Thus the median and lower and upper quartiles divide the PDF into four regions each containing
one quarter of the probability. Smaller subdivisions are also possible, e.g. the n-th percentile, P n ,
of a PDF is defined by F(P n ) = n/100.
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