.lst-kix_dhvd8yl62l91-8 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-8}.lst-kix_dhvd8yl62l91-0 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-0,decimal) ". "}.lst-kix_dhvd8yl62l91-1 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-1,lower-latin) ". "}.lst-kix_dhvd8yl62l91-5 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-5}ol.lst-kix_dhvd8yl62l91-3.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-3 0}.lst-kix_dhvd8yl62l91-2 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-2,lower-roman) ". "}.lst-kix_dhvd8yl62l91-3 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-3,decimal) ". "}.lst-kix_dhvd8yl62l91-4 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-4,lower-latin) ". "}.lst-kix_dhvd8yl62l91-5 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-5,lower-roman) ". "}ol.lst-kix_dhvd8yl62l91-5.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-5 0}.lst-kix_dhvd8yl62l91-2 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-2}.lst-kix_dhvd8yl62l91-8 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-8,lower-roman) ". "}ol.lst-kix_dhvd8yl62l91-1.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-1 0}.lst-kix_dhvd8yl62l91-6 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-6,decimal) ". "}.lst-kix_dhvd8yl62l91-7 > li:before{content:"" counter(lst-ctn-kix_dhvd8yl62l91-7,lower-latin) ". "}ol.lst-kix_dhvd8yl62l91-7.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-7 0}.lst-kix_dhvd8yl62l91-1 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-1}ol.lst-kix_dhvd8yl62l91-3{list-style-type:none}ol.lst-kix_dhvd8yl62l91-2{list-style-type:none}ol.lst-kix_dhvd8yl62l91-1{list-style-type:none}ol.lst-kix_dhvd8yl62l91-4.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-4 0}ol.lst-kix_dhvd8yl62l91-0{list-style-type:none}ol.lst-kix_dhvd8yl62l91-7{list-style-type:none}ol.lst-kix_dhvd8yl62l91-6{list-style-type:none}ol.lst-kix_dhvd8yl62l91-5{list-style-type:none}ol.lst-kix_dhvd8yl62l91-4{list-style-type:none}ol.lst-kix_dhvd8yl62l91-2.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-2 0}.lst-kix_dhvd8yl62l91-4 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-4}ol.lst-kix_dhvd8yl62l91-6.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-6 0}.lst-kix_dhvd8yl62l91-7 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-7}.lst-kix_dhvd8yl62l91-0 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-0}ol.lst-kix_dhvd8yl62l91-0.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-0 0}ol.lst-kix_dhvd8yl62l91-8{list-style-type:none}.lst-kix_dhvd8yl62l91-6 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-6}.lst-kix_dhvd8yl62l91-3 > li{counter-increment:lst-ctn-kix_dhvd8yl62l91-3}ol.lst-kix_dhvd8yl62l91-8.start{counter-reset:lst-ctn-kix_dhvd8yl62l91-8 0}
Mathematical expectation of a n.v. XIf you want to learn more check out What are the defining characteristics of coal mining?
Denoted by EX, given (when it exists)
By EX =
xPx(x)
=
CPx(x) = C
Rx(x) = C1 = CDon't forget about the age old question of What are the Berber influences from the North?
E(dx) = d E(x)If you want to learn more check out Describe the yellow river cultures.
E(dx) =
dxPx(x) = dEx
d ERWe also discuss several other topics like What is the Signal Detection Theory?
E(x + B) =
x + B Px(x)
=
xPx(x) + BPx(x) =
xPx(x) + B
Px(x)We also discuss several other topics like What is the difference between rote counting and rational counting?
E(x) + BDon't forget about the age old question of What are numbers calculated by comparing a number to a base number?
“Expectation is Linear”
E(dx +B) = dE(x) + B Vd, BER
In general it is not true that E(xy) = EXEY
Hint: to find a counterexample to the above equity, take x = y
Take X such that x = +1 w.p.
-1 w.p. 
Then E(x) = 1.
+ -1.
= 0
Take x = y then xy = x2 = 1
So E(xy) = E(x2) =1
EXEY = 0.0 = 0
Note: Let x be a n.v. and let f:R → R be a function
Then f(x) is also a n.v.

f(x) = f0x
So f0x is a function from
to R ie f0x is a n.v.
If it exists
Ef(x) =
f(x) P(x = x)
f(x) Px(x)
Existence for us is going to mean that
|f(x)|Px(x) < + 
Definition: let x be a n.v. if it exists, the variance of x denoted by varX is given by: varX = E[(x - Ex)2] = E(x2) - (EX)2
Example: Let X, Y, and Z be three distinct n.v. respectively given by
X = 0, y = +1 w.p.
-1 w.p.
and Z = +10 w.p.
-10 w.p. 
Then EX = 0, EY = 1.
+ (-1).
= 0, EZ = 10.
+ (-1).
) = 0
Var X = E[(E - EX)2] = E[x - 0]2 = E(x2) = 0 var Y = E[(y - EY)2] = E(12) = E(1) = 1
Var Z = E[(z - EZ)2] = E(102) = E(100) = 100
The higher the valence, the more spread out a random variable is vis a vis its expectation
Properties of variance
- If x = c, then var X = 0, indeed
Var X =
(x - EX)2 px(x)
=
(C - C)2 Px(x) =
OPx(x) = 0
- Var (∝X) =
(dx - E(dx)2Px(x)
= ∝2 var X
- Var (x + B) =
(x + B - E(x + B))2 Px(x)
(x + B - EX - B)2 Px(x)
= var X
- In general, var (x + y )
var X + var Y
Indeed taking x = y we have
var(X + Y) = var(2x) = 4 var X
var X + var Y = 2 var X
- Var X = 0 < = >
(x - EX)2 Px(x) = 0
So var x = 0 < = > x - EX = 0
Definition: the standard deviation of a n.v. X is the positive square root of the variance, of O3x denotes the variance of X, so 6x denotes the standard deviation of X, in particular O2x = |∝}Ox and 6x + B = 6x