LECTURE 5: Discrete random variables:

probability mass functions and expectations

• Random variables: the idea and the definition

- Discrete: take values in finite or countable set

• Probability mass function (PMF)

• Random variable examples

- Bernoulli

Uniform

Binomial

Geometric

• Expectation (mean) and its properties

The expected value rule

- Linearity

Random variables: the idea

e fo ma ·sm

• A andom variable ( u .v ") assoc·ates a value (a umbe ) to every possible outcome

• Mathematically: ,A fu ction fi om the sample spac ,e Q to the real numbers

• t can take discrete or continuous va ues

o a random variable X numer·cal value x

• We can have several random va ·ables defined on the same sample space

• A function of one o several random variables 1s a so a random variab l,e

- meaning of X + Y:

robabi ·ty mass · nction (PM ) of a discrete .v. X

• It is the uprobabil1ty law" or • probability distributio " of X

• If we t·x some x, then ux - x" ·s an event

Px(x) - P(X - x) = P( {w E Q s.t. X(w) = x})

• Properties: Px(x) > O Px(x) = 1 X

1 prob= -

4

X

X

5

3

M calcula ·on

• wo ro Is of a tetra edral die • Le every possib e ,ou · come ave probability 1/16

4

Y = Second 3 roll ------

2

1

1 2 3 4

X = First roll

pz(z) )~

z _ x Y

• repeat for a ll z: col ect a I possib e outcomes for which Z ·s equa l to z

- add t ei p obab i ·ti es

,._ ,

1 2 3 4 5 6 7 8 9 z

Th simples ra dam variable. ernou · wi p ram

X = 1, 0,

w. - . p

w.p. 1 - p

rpE[ ,1

• Models a trial that results ·n success/fa·1ure, Heads/Tails, etc.

• Indicator r.v. of a event A . IA 1 1f A oc r

e·nomial random var·able, param ters. o ·t·ve int ger n; p E [ , 1

• Experiment. n independent tosses of a co·n with P(Heads) = p

• Sample space. Set of sequences of H and T, of length n

• andom variable X. numb ,e of eads observed

• Mode l of: number of successes in a given number of independent trials

HHH

1- p 'fff

3

2

0

Px(k)

n 3 n - 10 n 100 - - -0.4 0.25 0.08

0.35 0.07

0.2 0.3 0.06

0.25 p - 0.5 0.15 p - 0.5 0.05 p - 0.5 ,....... --- ~ 0.04 -2., 0.2 C,

X X 0.. 0.. 0..

0.15 0.1 0.03

0.1 0.02 0.05

0.05 0.01

0 0 0 -1 -0.5 0 0.5 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 0 10 20 30 40 50 60 70 80 90 100

l{ X l{

0.7 0.35 0.14 ( )

0.6 0.3 0.12 ) -

p 0.2 p 0.2 G: p - 0.1 - - -0.5 0.25 0.1 G> -

G:

0.4 0.2 0.08 -'x 'x 'x )

'--"x x x 0.. 0.. 0..

0.3 0.15 0.06 Ci: -:i>

0.2 0.1 0.04 -Gl Ki:)

0.1 0.05 0.02 Ll ti> -

0 ~ 0 0 -

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 0 10 20 30 40 50 60 70 80 90 100 X X X

Geometric rando v, riab e: para e er p: o < p < 1

• Experiment. i finitely many independent toss ,es of a coin P( eads) - p

• Sample space: Set of inf 1 ·te seque ces of H and T

• Random variable X. numbe of tosses unt·1 the first Heads

• Model of: wa·ting times; number of tria s u til a success

Px(k) -

Px(k)

( o He d ev r) p-1/3

1 2 3 4 5 6 7 8 9 k

Exp ctation/mean o a random variable

• Mo ivation. Play a game 1000 times. Random gain at each play described by:

1, w.p. 2/10

X- 2, w.p. 5/10

4, w.p. 3/10

• "Average" gain:

• Definition [X] X

• In erpretation. Av ,erage in large number of indepe · dent repetitions oft e experiment

• Caut1 n. If we have an inf·n·te sum it needs to be well-defined.

We assu e ~]xi Px(x) < oo X

Expec a ·on of a Ber o Iii r.v

X= 1,

0, w.p. P

w.p 1 - p

If X is the indicator of an event A, X IA :

xpec ation of a ·rorm r v.

• nifor on0,1, ... ,n

Px(x) ~

1

n+l

0 1

E[X

••••••

n X

X

Expectation as a population average

• n students

• Weight of ith student: xi

• Exp ,eriment: pick a student at random, all equally likely

• Random variable X : weight of selected student

- assume the xi are distinct

Px(xi) =

E[X] =

le en ary properties of expec a ions

• Defini ·on: [X]

• If X > 0 , then E[X] > X

• If a < X < b, then a < [X] < b

• If c 1s a constant, E[c C

he expecte value rule, for calc ating E[g(X)]

• Let X be a r.v. and let Y - g(X)

• Averaging over y: E[Y] - LYPy(y) y

prob

• Averag·ng over x:

X

Proof:

X y

g

• Ca t10 · In g,e e al, E[g(X) -::/= g(E[X])

Linearity of expectation: E[aX + b] = aE[X] + b

• Intuitive

• Derivation, bas,ed on the expected value rule:

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Resource: Introduction to Probability John Tsitsiklis and Patrick Jaillet

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