Moments and Other Measures in Terms of Expectations
Data Science and A.I. Lecture Series
By Bindeshwar Singh Kushwaha – PostNetwork Academy
Moments
The \( r^{th} \) order moment about any point \( A \) of a variable \( X \) is given by:
For discrete variables:
\[ \mu_r’ = \sum_{i=1}^{n} p_i (x_i – A)^r \]
For continuous variables:
\[ \mu_r’ = \int_{-\infty}^{\infty} (x – A)^r f(x) dx \]
Central Moments
The \( r^{th} \) order central moment is given by:
For discrete variables:
\[ \mu_r = \sum_{i=1}^{n} p_i (x_i – \mu)^r \]
For continuous variables:
\[ \mu_r = \int_{-\infty}^{\infty} (x – \mu)^r f(x) dx \]
Expectation form:
\[ \mu_r = E[(X – \mu)^r] \]
Variance
Variance of a random variable \( X \) is the second-order central moment:
\[ V(X) = E[X^2] – (E[X])^2 \]
Using moments about the origin:
\[ V(X) = \mu_2′ – (\mu_1′)^2 \]
Theorem: Variance Scaling Property
If \( X \) is a random variable and \( a, b \) are constants, then:
\[ V(aX + b) = a^2 V(X) \]
Proof of Theorem
By definition of variance:
\[ V(aX + b) = E[(aX + b – E[aX + b])^2] \]
Expanding expectation:
\[ = E[(aX + b – aE[X] – b)^2] \]
Simplifying:
\[ = E[a^2 (X – E[X])^2] \]
Using expectation properties:
\[ = a^2 E[(X – E[X])^2] = a^2 V(X) \]
Example: Variance Calculation
| X |
p(X) |
| -2 |
0.15 |
| -1 |
0.30 |
| 0 |
0 |
| 1 |
0.30 |
| 2 |
0.25 |
Computing \( V(X) \):
\[ V(X) = E[X^2] – (E[X])^2 \]
Using values:
\[ V(X) = 2.2 – (0.2)^2 = 2.2 – 0.04 = 2.16 \]
Variance of a Linear Transformation
Computing \( V(2X + 3) \):
Using theorem: \( V(aX + b) = a^2 V(X) \)
\[ V(2X + 3) = 4V(X) \]
Substituting \( V(X) = 2.16 \):
\[ V(2X + 3) = 4(2.16) = 8.64 \]
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