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 X
X
• It is the uprobabil1ty law" or • probability distributio " of X
5
• If we t·x some x, then ux - x" ·s an event
1
prob= -
4
Px(x) - P(X - x) = P({w E Q s.t. X(w) = x})
• Properties: Px(x) >O Px(x) =1
X
M calcula ·on
• wo ro Is of a tetra edral die • Le every possib e ,ou · come ave probability 1/16
4
z _x Y
Y = Second 3 • repeat for a ll z:
roll ------
2 col ect a I possib e outcomes for which Z ·s equa l to z
- add t ei p obab i ·ti es
1
1 2 3 4
X = First roll
pz(z) )~
,._
,
1 2 3 4 5 6 7 8 9 z
Th simples ra dam variable. ernou · wi p ram rpE[ ,1
w. - . p
X = 1,
0, w.p. 1 - p
• 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
Px(k)
0
1- p 'fff
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
,.......
-2.,
X 0.2 ---
C,
X ~ 0.04
0.. 0.. 0..
0.1
0.15 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 0.12 -
-
0.3 )
0.5
p - 0.2
0.25
p 0.2
0.1
G:
G>
p 0.1
-
G:
0.4 0.2 0.08 -
)
'x 'x 'x
'--"x
0..
x
0..
x
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)
p-1/3
( o He d ev r)
1 2 3 4 5 6 7 8 9 k
Exp ctation/mean o a random variable
1, w.p. 2/10
• Mo ivation. Play a game 1000 times.
X- 2, w.p. 5/10
Random gain at each play described by:
4, w.p. 3/10
• "Average" gain:
[X] • In erpretation. Av ,erage in large number
• Definition
X 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
1, w.p. P
X=
0, 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 n X
E[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]
X
• If X > 0, then E[X] >
• If a < X < b, then a < [X] < b
• If c 1sa constant, E[c C
he expecte value rule, for calc ating E[g(X)] X y
• Let X be a r.v. and let Y - g(X) g
• Averaging over y: E[Y] - LYPy(y)
y
prob
• Averag·ng over x:
Proof:
• 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:
MIT OpenCourseWare
https://ocw.mit.edu
Resource: Introduction to Probability
John Tsitsiklis and Patrick Jaillet
The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource.
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms.
