Download Equally Likely Outcomes

CS723 - Probability
and
Stochastic Processes
Lecture No. 09
In Previous Lectures
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Probability theory
Equally likely outcomes
Thoroughly analyzed the random partitioning
of a unit interval with two randomly chosen
points
The points were equally likely to fall on any
point on unit interval
Equally Likely Outcomes
Coin toss: two equally likely outcomes
Dice roll: six equally likely outcomes
Semi-cube roll: not equally likely, so
mark the
sides A, B, C, D, E, and F
Sack with 6 green and 3 red balls: probability of
2 green balls =41.67% probability of 2 red balls
= 8.33%
Number of customers entering in a restaurant in a
10-minute interval.
Random Variables
Transformation or mapping of outcomes to
numbers on real line
Probability maps values of RV to [0,1]
Roll of two dice: X is random variable
that
represents the sum
Pr(X=2) = Pr({(1,1)}) = f(2) = 1/36
f(4) = Pr(X=4) = Pr({(3,1),(2,2),(1,3)}) = 3/36
Probability Mass Function
6/36
5/36
4/36
3/36
2/36
1/36
2
4
6
8
Value of the random variable X
10
12
Chuk-a-luck
Three dice are thrown and you pick a
number
between 1 and 6
You loose one dollar if your number does
not
show up on any dice
You win 3 dollars if your number shows up on all
three dice
X is your gain/loss and Δ = {-1, 1, 2, 3}
f(3) = 1/216, f(2) = 15/216 = 5/72,
f(1) = 75/216 = 25/72 & f(-1)= 125/216
Binomial Distribution
Sample space with only two outcomes success
and failure
Pr(success) = p & Pr(failure) = (1-p)
Repeat the experiment N times under same
conditions (with replacement)
Total number of outcomes in sample space is 2N
(words from alphabets)
Random variable is X = No. of successes
f(k) = Pr(X=k) = (N,k) pk (1-p)N-k
People Entering KFC
When a person enter the franchise, he enters
instantaneously
People enter the franchise independently
of
each other
The time is divided in 5 secs intervals
0.1
0.05
0
5
10
15
20
25
30
35
(Binomial) Number of people entering KFC in 10 minutes
40
45
50
Poisson Distribution
Probability of success VERY small and
number of repetitions VERY large
The random variable can take any value
that is a positive integer (or zero)
PMF is given by f(k) = Pr(X=k) = e-λ(λk/k!)
0.1
0.05
0
5
10
15
20
25
30
35
(Poisson) Number of people entering KFC in 10 minutes
40
45
50
Cumulative Distribution
Function
CDF is continuous on the real line and is
probability of a complex event
F(t) = Pr(-∞ < X ≤ t) = ∑ f(a) s.t. a ≤ t, t ε R
lim t → -∞ F(t) = 0 and lim t → ∞ F(t) = 1
f(t) = F(t) – lim s → t F(s)
Important Points
A random variable is a function from sample
space to real line
Finite sample spaces always give discrete
random variables
Probabilities are assigned to events in terms of
values of random variable
We can work with a PMF without knowing the
underlying experiment
Sum of all values of PMF should be 1