Expectation vs AVerage (AM)
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?
Weighted Mean Revisited
400 |
70 |
0.56 |
28000 |
224 |
500 |
35 |
0.28 |
17500 |
140 |
800 |
20 |
0.16 |
16000 |
128 |
Total |
125 |
1 |
61500 |
492 |
If a man is randomly selected, what is the probability that his daily
income is 400 BDT?
- \(AM =
\frac{61500}{125}=492\)
- In expectation, \(rf\) is the weight
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 \(\Sigma f_i=N\))
- \(=
x_1\frac{f_1}{N}+x_2\frac{f_2}{N}+\cdots +x_2\frac{f_n}{N}\)
- \(=
x_1p_1+x_2p_2+\cdots+x_np_n\)
- \(= \sum x_i\cdot
p(x_i)\)
Expectation
Probabilities are weights
\[\begin{eqnarray}
E(X) &=& x_1 \times P(x_1) + x_2 \times P(x_2)+ \cdots +x_n
\times P(x_n) \nonumber \\
&=& \displaystyle \sum_{i=1}^n x_i \times P(x_i) \nonumber \\
\end{eqnarray}\]
For continuous X
\(\displaystyle E(X) =
\int_{-\infty}^{\infty} f(x)dx\)
- \(E(X^2) = \sum x^2
\times p(x)\)
Expectation Example
\(P(x)\) |
\(\frac 1
4\) |
\(\frac 1
2\) |
\(\frac 1
4\) |
Find
- \(E(X)\)
- \(E(X^2)\)
Properties of Expectation
Aslo
refer to the properties of AM (\(E(X)=\bar
X\))
- Expectation of a constant, \(E(c)=c\)
- \(E(aX)=aE(X)\)
- \(E(X-a) = E(X) -
E(a)\)
- \(E(aX+b) =
aE(X)+b\)
- \(E(X+Y) =
E(X)+E(Y)\)
- \(E(XY) = E(X)
E(Y)\) (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?
- \(V(X+a)=V(X)\)
- \(V(aX+c) =
a^2V(X)\) (depends only on scale, recall)
- \(V(X+Y) =
V(X)+V(Y)\)
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)
- \(E(XY)-E(X)E(Y)\)
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
Double Summation Revisited
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(X2) ≥ {E(X)}2
\(V(X)\ge 0\)
- \(\therefore E(X^2) -
\{E(X)\}^2 \ge 0\)
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 \(0.5 \gt \frac 1 3
\to 3 \gt \frac 1 {0.5}\)
Variance of constant (a)
\(V(X) = E\{X-E(X)\}^2\)
- \(V(a) =
E\{a-E(a)\}^2\)
- \(E(a-a)^2\)
- \(E(0)=0\)
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?
- \(V(X+Y)=?\)
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)
- Find k
- 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
- k
- 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
- k
- V(2x-1)
Example 05: y = 3x+5. Find V(3y-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