14
Stanley Chan 2020. All Rights Reserved. ECE 302: Lecture 3.7 Binomial Random Variables Prof Stanley Chan School of Electrical and Computer Engineering Purdue University 1 / 14
©Stanley Chan 2020. All Rights Reserved.
ECE 302: Lecture 3.7 Binomial Random Variables
Prof Stanley Chan
School of Electrical and Computer EngineeringPurdue University
1 / 14
©Stanley Chan 2020. All Rights Reserved.
Data coming from a binary image sensor
In 2005, a new type of image sensor was proposed. The sensor is calledthe Quanta Image Sensor.
Every pixel is binary: Either 1 or 0. Probability of getting a 1 is p.The sensor can buy you...
2 / 14
©Stanley Chan 2020. All Rights Reserved.
The power of quanta image sensorsAbhiram Gnanasambandam, Stanley H. Chan, “Image Classification in the Dark using Quanta Image Sensors” EuropeanConference on Computer Vision (ECCV 2020)
3 / 14
©Stanley Chan 2020. All Rights Reserved.
The power of quanta image sensors
One basic question is:
I have observed 100 frames.
Since the pixels are binary, I can count the number of 1’s and 0’s foreach pixel.
What is the statistics of these 1’s and 0’s?4 / 14
©Stanley Chan 2020. All Rights Reserved.
Outline
3.1 Random variables
3.2 Probability mass functions (PMF)
3.3 Cumulative distribution functions (discrete case)
3.4 Expectation
3.5 Moments and variance
3.6 Bernoulli random variables
3.7 Binomial random variables
Definition of binomial random variablesRelationship with BernoulliExpectation and varianceApplication: Binary image sensors
3.8 Geometric random variables
3.9 Poisson random variables
5 / 14
©Stanley Chan 2020. All Rights Reserved.
Binomial Random Variable
Definition
Let X be a Binomial random variable. Then, the PMF of X is
pX (k) =
(n
k
)pk(1− p)n−k , k = 0, 1, . . . , n,
where 0 < p < 1 is the Binomial parameter, and n is the total number ofstates. We write
X ∼ Binomial(n, p)
to say that X is drawn from a Binomial distribution with a parameter p ofsize n.
Example. Number of heads in n coin flips.
6 / 14
©Stanley Chan 2020. All Rights Reserved.
Origin of binomial random variables
Flip a coin 3 times. Find the probability of getting 3 heads.
pX (3) = P[{“HHH”}] = P[{“H”} ∩ {“H”} ∩ {“H”}](a)= P[{“H”}]P[{“H”}]P[{“H”}] (b)
= p3,
Find the probability of getting 2 heads.
pX (2) = P[{“HHT”} ∪ {“HTH”} ∪ {“THH”}](c)= P[{“HHT”}] + P[{“HTH”}] + P[{“THH”}]= p2(1− p) + p2(1− p) + p2(1− p) = 3p2(1− p),
7 / 14
©Stanley Chan 2020. All Rights Reserved.
Origin of binomial random variables
In general,
pX (k) =
(n
k
)︸ ︷︷ ︸
number of combinations
pk︸ ︷︷ ︸prob getting k H’s
(1− p)n−k︸ ︷︷ ︸prob getting n − k T’s
. (1)
8 / 14
©Stanley Chan 2020. All Rights Reserved.
Shape of a binomial PMF
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50 60
p = 0.1
p = 0.5
p = 0.9
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60
n = 5
n = 50
n = 100
(a) n = 60 (b) p = 0.5
Table: PMFs of a binomial random variable X ∼ Binomial(n, p).
9 / 14
©Stanley Chan 2020. All Rights Reserved.
Moments of Binomial
Property
If X ∼ Binomial(n, p), then
E[X ] = np,
E[X 2] = np(np + (1− p)),
Var[X ] = np(1− p).
Proof.
E[X ] =n∑
k=0
k ·(n
k
)pk(1− p)n−k =
n∑k=0
k · n!
k!(n − k)!pk(1− p)n−k
=n∑
k=1
n!
(k − 1)!(n − k)!pk(1− p)n−k .
... a few more steps.10 / 14
©Stanley Chan 2020. All Rights Reserved.
A short cut to the proof
11 / 14
©Stanley Chan 2020. All Rights Reserved.
PMF and CDF
FX (k) =k∑
`=0
(k
`
)p`(1− p)k−`. (2)
0
0.05
0.1
0.15
0.2
0 5 10 15 20 25 30 0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
Table: PMF and CDF of a binomial random variable X ∼ Binomial(n, p).
12 / 14
©Stanley Chan 2020. All Rights Reserved.
Going back to the binary sensor...
How to model the random variable X = number of 1’s observed in 100measurements?
13 / 14
©Stanley Chan 2020. All Rights Reserved.
Questions?
14 / 14
