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Get Full Access to GATech - MATH 3670 - Class Notes - Week 4
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GATECH / Math / MATH 3670 / What does PMF stand for?

# What does PMF stand for? Description

##### Description: dealing with PMF (probability mass function) and CDF (cumulative distribution function)
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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:

1.  > {H,T}

x(H) = -10If you want to learn more check out What is the Signal Detection Theory?

x(T) = 20

1.  = {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

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