Probability
Definitions
Random experiment: an experiment with a variety of possible outcomes eg toss a coin
Sample space: set of possible outcomes in experiment eg { Heads , Tails }
Event: subset of the sample space eg “Heads”: { Heads }
The probability of some outcome measures how likely it is to occur: the proportion of times it
would happen if the experiment were repeated many times
Random Variables
Random variable is a numerical variable whose value depends on the outcome of a random
experiment
The support of a random variable is the set of values it can take
Discrete Random Variables Support
Z= −1 if Tails
{ {−1,1 }
+1 if Heads
X =Number on die { 1,2,3,4,5,6 }
Y = 1 if number on die is odd
{ { 0,1 }
0 otherwise
Probability Distribution of a Discrete Random Variable
If X is a random variable with support { x 1 , x 2 , … , x n }, its probability distribution is a list of
probabilities, one for each value: f X ( x 1 ) , f X ( x 2 ) , … , f X ( x n )
f X ( ⋅ ) is the probability mass function (PMF)
n
Note that: P ( X =x1∨ X=x 2 )=f X ( x 1 ) + f X ( x 2 ); Σ i=1 f X ( x i )=1
Cumulative Probability Distribution Function
The CDF of random variable X defined as F X ( x )=P ( X ≤ x ) , x ∈ [ −∞ ,+∞ ]
So if X has support { x 1 , x 2 , … , x n } and PMF f X ( ⋅) ⇒ F X ( x ) =Σ x ≤x f X ( xi )
i
F X ( ⋅ ) is an increasing function; F X ( - ∞ )=0 , F X ( ∞ )=1
Examples of Discrete Distributions
Discrete Uniform Distribution (with parameter n)
1
o Random variable N with support { 1,2 , … , n } ⇒ f N ( i )= , i∈ [ 1 , … , n ]
n
o Eg fair die, n=6
Two point Distribution
o Random variable X with support { l , h } ⇒ f X ( l ) =1− p ; f X ( h )= p
Definitions
Random experiment: an experiment with a variety of possible outcomes eg toss a coin
Sample space: set of possible outcomes in experiment eg { Heads , Tails }
Event: subset of the sample space eg “Heads”: { Heads }
The probability of some outcome measures how likely it is to occur: the proportion of times it
would happen if the experiment were repeated many times
Random Variables
Random variable is a numerical variable whose value depends on the outcome of a random
experiment
The support of a random variable is the set of values it can take
Discrete Random Variables Support
Z= −1 if Tails
{ {−1,1 }
+1 if Heads
X =Number on die { 1,2,3,4,5,6 }
Y = 1 if number on die is odd
{ { 0,1 }
0 otherwise
Probability Distribution of a Discrete Random Variable
If X is a random variable with support { x 1 , x 2 , … , x n }, its probability distribution is a list of
probabilities, one for each value: f X ( x 1 ) , f X ( x 2 ) , … , f X ( x n )
f X ( ⋅ ) is the probability mass function (PMF)
n
Note that: P ( X =x1∨ X=x 2 )=f X ( x 1 ) + f X ( x 2 ); Σ i=1 f X ( x i )=1
Cumulative Probability Distribution Function
The CDF of random variable X defined as F X ( x )=P ( X ≤ x ) , x ∈ [ −∞ ,+∞ ]
So if X has support { x 1 , x 2 , … , x n } and PMF f X ( ⋅) ⇒ F X ( x ) =Σ x ≤x f X ( xi )
i
F X ( ⋅ ) is an increasing function; F X ( - ∞ )=0 , F X ( ∞ )=1
Examples of Discrete Distributions
Discrete Uniform Distribution (with parameter n)
1
o Random variable N with support { 1,2 , … , n } ⇒ f N ( i )= , i∈ [ 1 , … , n ]
n
o Eg fair die, n=6
Two point Distribution
o Random variable X with support { l , h } ⇒ f X ( l ) =1− p ; f X ( h )= p
