Mathematical Expectation

Abdullah Al Mahmud

Expectation vs AVerage (AM)

Accident	Frequency (f)	Relative Frequency (rf)
Bus	70	0.56
Train	35	0.28
Ship	20	0.16
Total	125	1

• Do rfs look like probabilities?
• What is the probability that an accident is ocurred in a Bus service?

Expectation from Mean

\(\displaystyle \bar X=\frac{\Sigma x_if_i}{\Sigma f_i}\)

• \(= \frac{x_1f_1+x_2f_2+\cdots+x_3f_3}{N}\) (letting \(\Sigma f_i=N\))
• \(= x_1\frac{f_1}{N}+x_2\frac{f_2}{N}+\cdots +x_2\frac{f_n}{N}\)
• \(= x_1p_1+x_2p_2+\cdots+x_np_n\)
• \(= \sum x_i\cdot p(x_i)\)

\(E(X^2)\)

• \(E(X^2) = \sum x^2 \times p(x)\)
• Why not \(\sum x^2 \times p(x^2)\)
• Say x = 0, 1 and P(x) = 1/2, 1/2
• What are \(P(x^2)\)
• Make a table to feel

Expectation Example

\(X\)	0	1	2
\(P(x)\)	\(\frac 1 4\)	\(\frac 1 2\)	\(\frac 1 4\)

Find

1. \(E(X)\)
2. \(E(X^2)\)

Properties of Expectation

• Expectation of a constant, \(E(c)=c\)
• \(E(aX)=aE(X)\)
• \(E(X \pm a) = E(X) \pm E(a)\)
• \(E(aX \pm b) = aE(X) \pm b\)
• \(E(X \pm Y) = E(X) \pm E(Y)\)
• \(E(XY) = E(X) E(Y)\) (if independent, relate with probability)
• \(E(X^2)\ge \{E(X)\}^2\)
• \(E\left(\frac 1 X \right) \ge \frac 1 {E(X)}\)

Variance

Recall this (click)

We knew, \(\sigma^2 = \bar {X^2} - (\bar X)^2=\) Mean of square - square of mean

• \(V(X)=E(X^2)-\{E(X)\}^2\)
• Original Formula: \(V(X) = E\{x-E(X)\}^2\) (Match)
• Expand and prove these are equal

Variance Properties

• \(V(c)=0\) ponder, why?
• \(V(X \pm a)=V(X)\)
• \(V(aX \pm c) = a^2V(X)\) (depends only on scale, recall)
• \(V(X \pm Y) = V(X) \pm V(Y)\) (if independent)
• \(V(X \pm Y) = V(X) \pm V(Y) + 2 Cov(X,Y)\) (if not independent)

Covariance

\(Cov(X,Y) = \frac{\Sigma (x-\bar x)(y-\bar x)}{n}\)

Write in terms of \(E(X)\)

• \(Cov(X,Y) = E [\{x-E(X)\}\{y-E(Y)\}]\) (Expand)
• \(E(XY)-E(X)E(Y)\)

Covariance for Independent (X, Y)

\(Cov(X,Y) = E(XY)-E(X)E(Y)\) = 0

So what do we get?

Prove E(X) Properties

Go back to see the properties

E(a)

\(E(a) = \sum a \cdot p(x_i)\)

• \(=a \cdot \sum p(x_i)\)
• \(=a \times 1 = a\)
• Others can be proven similarly

E(aX)

\(E(aX) = \sum a x_i \cdot p(x_i)\)

• \(=a \sum x_i \cdot p(x_i)\)
• \(=a \times E(X) = a E(X)\)
• Others can be proven similarly

Double Summation Revisited

Exam (X) \(\to\) Result (Y) \(\downarrow\)	PSC	JSC	SSC	HSC	Total
Passed	30	26	23	25	104
Failed	12	13	10	14	49
Absent	5	2	3	4	14
Total	47	41	36	43	167

• If sum over \(x_i\), we get only 1 column at a time
• If sum over \(y_i\), we get only 1 row at a time
• SO to get grand total, we must use \(\sum \sum (x_i+y_j)\)

E(X+Y) & E(XY)

\(E(X+Y)\)

• \(= \displaystyle \sum_{i=1}^m \sum_{j=1}^n (x_i+y_j)P(x_i,y_j)\)
• \(=\displaystyle \sum_{i=1}^m \sum_{j=1}^n \{x_iP(x_i,y_j)+y_jP(x_i,y_j)\}\)
• \(=\displaystyle \sum_{i=1}^m \sum_{j=1}^n x_iP(x_i,y_j)+\sum_{i=1}^m \sum_{j=1}^n y_jP(x_i,y_j)\)
• \(=\displaystyle \sum_{i=1}^m x_i\sum_{j=1}^n P(x_i,y_j)+\sum_{j=1}^n x_i\sum_{i=1}^m P(x_i,y_j)\)
• \(=\displaystyle \sum_{i=1}^m x_i P(x_i) + \sum_{j=1}^n y_j P(y_j)\)
• \(=E(X)+E(Y)\)

E(XY)

\(\displaystyle E(XY) = \sum_{i=1}^m \sum_{j=1}^n (x_iy_j)P(x_i,y_j)\)

• \(\displaystyle \sum_{i=1}^m \sum_{j=1}^n (x_iy_j)P(x_i) P(y_j)\) (independent)
• \(\displaystyle \{\sum_{i=1}^m x_i P(x_i) \}\{\sum_{j=1}^n y_j P(y_j)\}\)
• \(E(X) E(Y)\)

E(X²) ≥ {E(X)}²

\(V(X)\ge 0\)

• \(E(X^2) - \{E(X)\}^2 \ge 0\)
• E(X²) ≥ {E(X)}²

AM & HM

\(E(\frac 1 X) \ge \frac 1 {E(X)}\)

• \(AM \ge HM\)
• \(HM =\) opposite of mean of opposite \(\to \frac{1}{\frac{\sum \frac 1 {x_i}}{n}} = \frac 1 {E(\frac 1 x)}\)
• \(E(X) \ge \frac{1}{E(\frac 1 x)}\)
• If \(a \gt b \to \frac 1a \lt \frac 1b\)

Variance of constant (a)

\(V(X) = E\{X-E(X)\}^2\)

• \(V(a) = E\{a-E(a)\}^2\)
• \(E(a-a)^2\)
• \(E(0)=0\)
• Logically why?

V(aX+b)

\(V(X) = E\{X-E(X)\}^2\)

\(V(aX+b)=\)

• \(E\{(aX+b)-E(aX+b)\}^2\)
• \(E\{aX+b-aE(X)-b\}^2\)
• \(E\{aX-aE(X)\}^2\)
• \(a^2E\{X-E(X)\}^2\)
• \(a^2V(X)\)
• Does variance depend on origin and scale?

V(X+Y)

For independent variables

• \(V(X+Y) = E\{(X+Y) - E(X+Y)\}^2\)
• \(E\{(X+Y - E(X) - E(Y) \}^2\)
• \(E[\{X-E(X)\} + \{ Y- E(Y)\}]^2\)
• Expand \((a+b)^2\) formula
• V(X) + 2Cov(X,Y) + V(Y)
• V(X) + V(Y)

Cov(X,Y) Properties

For independent X, Y

• \(\to E(XY) = E(X)E(Y)\)
• \(Cov(X,Y) = E(XY) - E(X)E(Y) = 0\)
• Correlation, \(r = \frac{Cov(X,Y)}{\sigma_x \sigma_y}=0\)

Example 01: E(X) & V(X)

x	-3	-2	-1	0	1	2
P(x)	k	2k	3k	2k	4k	0.4

1. Find k
2. Find \(E(X)\) and \(V(X)\)

Example 02: \(P(x) = \frac 1 n\)

\(P(x) = \frac 1 n; x = 1,2,3, \cdots,n\) Find \(E(X)\) and \(V(X)\)

Example 03

\(P(x)=\frac {3-|4-x|} k; x = 2,3,4,5,6\)

Find

1. k
2. V(2X-1)

Example 04: \(P(x) = \frac{3-|4-x|}{k}\)

\(P(x) = \frac{3-|4-x|}{k}; x = 2, 3, 4, 5, 6\)

Find

1. k
2. E(2X-3)
3. V(5x+6)

Y = 3X + 5

Find V(3y-2)

x	-2	-1	0	1	2
P(x)	0.20	0.15	0.10	0.15	0.40

• V(3y-2) = \(3^2V(y)\)
• \(9V(3x+5)\)
• \(9 \cdot 3^2 \cdot V(x)\)

Example 06: \(f(x) = \frac x 8\)

\(f(x) = \frac x 8; 0 <x<4\)

Find

• \(E(X)\)
• \(E(2x^2+3)\)
• \(V(X)\)

Word problems

Ball Selection

A pot contains 5 white and 8 black balls. If x denotes the number of black balls drawn, find E(x) and V(x).

• \(P(2 Black, 1 White) = \frac{8C_3 \times 5C_1}{13C_3}\)
• In the problem, x = 0, 1, 2, 3
• \(P(x) = \frac{8c_x \times ?}{13C_3}\)

Rain Coat Expectation

If it rains, a dealer earns 5,000 tk a day, and if it does not, he loses 1000 tk a day. Find his expected earning.