.lst-kix_ouhwessnt4da-8 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-8,lower-roman) ". "}ol.lst-kix_ouhwessnt4da-0.start{counter-reset:lst-ctn-kix_ouhwessnt4da-0 0}.lst-kix_e5uvguyfawll-8 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-8}ol.lst-kix_ouhwessnt4da-3.start{counter-reset:lst-ctn-kix_ouhwessnt4da-3 0}.lst-kix_ouhwessnt4da-7 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-7}.lst-kix_ouhwessnt4da-4 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-4,lower-latin) ". "}.lst-kix_ouhwessnt4da-3 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-3,decimal) ". "}ol.lst-kix_ouhwessnt4da-7.start{counter-reset:lst-ctn-kix_ouhwessnt4da-7 0}.lst-kix_e5uvguyfawll-3 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-3}.lst-kix_ouhwessnt4da-5 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-5,lower-roman) ". "}.lst-kix_ouhwessnt4da-6 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-6,decimal) ". "}.lst-kix_ouhwessnt4da-1 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-1}ol.lst-kix_e5uvguyfawll-2.start{counter-reset:lst-ctn-kix_e5uvguyfawll-2 0}.lst-kix_ouhwessnt4da-7 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-7,lower-latin) ". "}ol.lst-kix_e5uvguyfawll-3.start{counter-reset:lst-ctn-kix_e5uvguyfawll-3 0}ol.lst-kix_ouhwessnt4da-4.start{counter-reset:lst-ctn-kix_ouhwessnt4da-4 0}.lst-kix_ouhwessnt4da-8 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-8}.lst-kix_ouhwessnt4da-2 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-2}.lst-kix_e5uvguyfawll-7 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-7,lower-roman) ". "}.lst-kix_ouhwessnt4da-5 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-5}.lst-kix_e5uvguyfawll-8 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-8,decimal) ". "}.lst-kix_ouhwessnt4da-2 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-2,lower-roman) ". "}ol.lst-kix_e5uvguyfawll-6.start{counter-reset:lst-ctn-kix_e5uvguyfawll-6 0}.lst-kix_e5uvguyfawll-1 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-1,lower-roman) ". "}.lst-kix_e5uvguyfawll-3 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-3,lower-latin) ". "}.lst-kix_ouhwessnt4da-1 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-1,lower-latin) ". "}.lst-kix_e5uvguyfawll-2 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-2,decimal) ". "}.lst-kix_e5uvguyfawll-6 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-6,lower-latin) ". "}.lst-kix_e5uvguyfawll-5 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-5,decimal) ". "}.lst-kix_e5uvguyfawll-5 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-5}.lst-kix_ouhwessnt4da-0 > li:before{content:"" counter(lst-ctn-kix_ouhwessnt4da-0,decimal) ". "}.lst-kix_e5uvguyfawll-2 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-2}ol.lst-kix_ouhwessnt4da-1.start{counter-reset:lst-ctn-kix_ouhwessnt4da-1 0}.lst-kix_e5uvguyfawll-4 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-4,lower-roman) ". "}ol.lst-kix_e5uvguyfawll-2{list-style-type:none}ol.lst-kix_e5uvguyfawll-1{list-style-type:none}ol.lst-kix_e5uvguyfawll-4{list-style-type:none}ol.lst-kix_ouhwessnt4da-5.start{counter-reset:lst-ctn-kix_ouhwessnt4da-5 0}ol.lst-kix_e5uvguyfawll-3{list-style-type:none}.lst-kix_ouhwessnt4da-4 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-4}ol.lst-kix_e5uvguyfawll-0{list-style-type:none}ol.lst-kix_e5uvguyfawll-4.start{counter-reset:lst-ctn-kix_e5uvguyfawll-4 0}.lst-kix_e5uvguyfawll-0 > li:before{content:"" counter(lst-ctn-kix_e5uvguyfawll-0,lower-latin) ". "}ol.lst-kix_e5uvguyfawll-0.start{counter-reset:lst-ctn-kix_e5uvguyfawll-0 0}.lst-kix_e5uvguyfawll-6 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-6}ol.lst-kix_ouhwessnt4da-8.start{counter-reset:lst-ctn-kix_ouhwessnt4da-8 0}ol.lst-kix_ouhwessnt4da-2.start{counter-reset:lst-ctn-kix_ouhwessnt4da-2 0}.lst-kix_e5uvguyfawll-0 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-0}ol.lst-kix_e5uvguyfawll-7.start{counter-reset:lst-ctn-kix_e5uvguyfawll-7 0}.lst-kix_e5uvguyfawll-1 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-1}.lst-kix_e5uvguyfawll-7 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-7}ol.lst-kix_e5uvguyfawll-8.start{counter-reset:lst-ctn-kix_e5uvguyfawll-8 0}.lst-kix_e5uvguyfawll-4 > li{counter-increment:lst-ctn-kix_e5uvguyfawll-4}ol.lst-kix_e5uvguyfawll-5.start{counter-reset:lst-ctn-kix_e5uvguyfawll-5 0}.lst-kix_ouhwessnt4da-3 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-3}.lst-kix_ouhwessnt4da-6 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-6}.lst-kix_ouhwessnt4da-0 > li{counter-increment:lst-ctn-kix_ouhwessnt4da-0}ol.lst-kix_ouhwessnt4da-8{list-style-type:none}ol.lst-kix_ouhwessnt4da-7{list-style-type:none}ol.lst-kix_ouhwessnt4da-6{list-style-type:none}ol.lst-kix_ouhwessnt4da-5{list-style-type:none}ol.lst-kix_e5uvguyfawll-1.start{counter-reset:lst-ctn-kix_e5uvguyfawll-1 0}ol.lst-kix_ouhwessnt4da-4{list-style-type:none}ol.lst-kix_ouhwessnt4da-3{list-style-type:none}ol.lst-kix_ouhwessnt4da-2{list-style-type:none}ol.lst-kix_ouhwessnt4da-1{list-style-type:none}ol.lst-kix_ouhwessnt4da-0{list-style-type:none}ol.lst-kix_ouhwessnt4da-6.start{counter-reset:lst-ctn-kix_ouhwessnt4da-6 0}ol.lst-kix_e5uvguyfawll-6{list-style-type:none}ol.lst-kix_e5uvguyfawll-5{list-style-type:none}ol.lst-kix_e5uvguyfawll-8{list-style-type:none}ol.lst-kix_e5uvguyfawll-7{list-style-type:none}
Random VariablesWe also discuss several other topics like What are the properties of variance?
Definition: a random variable usually by x,y,z is a function with domain 
and co - domain RDon't forget about the age old question of When did the Haymarket Riot happen?
If the co - domain is Rd, d > 2, instead of R we have a random variable
Don't forget about the age old question of Who are the people of Sanhaja?
Function has the very precise definition
Vwe, F a unique x ER such that x(w) = x
Don't forget about the age old question of Describe the yellow river cultures.
Example:
-
> {H,T}
x(H) = -10If you want to learn more check out What is the Signal Detection Theory?
x(T) = 20
-
= {1,2,3,4,5,6}
Y: → RDon't forget about the age old question of What are preskills related to counting beyond 30?
I → y(i( = 2;
So y(i) = 2 y(2) = 4...etc
Z: = N → R
H → 2(k) = (-2)k
W: = [0,1] x [0,1] → R
(x,y) → w(x,y) = 
Recall that the range of a function x defined by R(x) is the set of all the x(W),wE,, sometimes repeated as x/()
For example at R(x) = {-10,20}
For y R(x) = {2,4,6,8,12}
Rz = {-2,4,-8,16…} = {(-1)x22: kEN)
Rw = [0,
]
Definition: called discrete if the range is finite on infinite constable
X: discrete
Y: discrete
Z: discrete (infinitely countable)
W: not discrete (infinitely many between 0 and )
PMF
Definition: the probability mass function (pmf) of the n.v. X, denoted by p, is given by p(x) = P({x = x}) = P(x = x) Vx ER(x)
Recall {x = x} = {wE: X(w) = x}
Let us take x from previous and y from previous
X: {H,T} → {-10,20}
Px(-10) = P(x = -10) = P({H})
Px(x) = P(x = 20) = P({T})
For y: = {1,2,...,6} → {2,4,...,12}
Py(2) = P(y = 2) = P({1})
Py(12) = P(y = 12) = P({12})
Example: let = {1,2,3,4,5,6}
P({i}) = 
Vie let
V: → R
I → V(i) = {i if i E {1,2,3} 6 - i if i E{4,5,6}
Find the pmf of V:
Solution:
V taake the values of 0,1,2,3
R(v) = {0,1,2,3}
Pv(0) = P(v = 0) = P({6}) =
Pv(1) = P(v = 1) = P({1,5}) = P({1}) + P({5}) = +=
Pv(2) = P(v = 2) = P({2,4}) = P(({2}) + P({4}) = ++
Pv(3) = P(v = 3) = P({3}) =
Definition: the cumulative distribution function (cdf) of a n,v, x, denoted by F or Fx, is given by f(x) = P({x<x}) = P(x<x) V x ER
Example: let = {1,2,...,6} with P({i}) =
And let V be as above / previous (range of V is {0,1,2,3}
Fv(x) = P(V < x)
F(-10) = 0 | F(0) =
F(-2) = 0 | F(
) =
F(-2) = 0 | F(
) =
f(i) = P(V<1)
= P(V = 0) + P(V = 1)
+=
Fv(x) = 0 if X < 0
if 0 < x < 1
if 1 < x < 2
if 2 < x < 3
1 if x > 3
