1 discrete math cs 2800 prof. bart selman [email protected] module probability --- part d) 1)...
Post on 20-Dec-2015
222 views
TRANSCRIPT
![Page 1: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/1.jpg)
1
Discrete MathCS 2800
Prof. Bart [email protected]
Module Probability --- Part d)
1) Probability Distributions2) Markov and Chebyshev Bounds
![Page 2: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/2.jpg)
2
Discrete Random variable
Discrete random variable– Takes on one of a finite (or at least countable) number of
different values.
– X = 1 if heads, 0 if tails
– Y = 1 if male, 0 if female (phone survey)
– Z = # of spots on face of thrown die
![Page 3: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/3.jpg)
3
Continuous Random variable
Continuous random variable (r.v.)– Takes on one in an infinite range of different values– W = % GDP grows (shrinks?) this year– V = hours until light bulb fails
For a discrete r.v., we have Prob(X=x), i.e., the probability that
r.v. X takes on a given value x.
What is the probability that a continuous r.v. takes on a specific value? E.g. Prob(X_light_bulb_fails = 3.14159265 hrs) = ??
However, ranges of values can have non-zero probability.
E.g. Prob(3 hrs <= X_light_bulb_fails <= 4 hrs) = 0.1
– Ranges of values have a probability
0
![Page 4: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/4.jpg)
4
Probability Distribution
The probability distribution is a complete probabilistic description of a random variable.
All other statistical concepts (expectation, variance, etc) are derived from it.
Once we know the probability distribution of a random variable, we know everything we can learn about it from statistics.
![Page 5: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/5.jpg)
5
Probability Distribution
Probability function– One form the probability distribution of a discrete
random variable may be expressed in.
– Expresses the probability that X takes the value x as a function of x (as we saw before):
)( xXPxPX
![Page 6: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/6.jpg)
6
Probability Distribution
The probability function – May be tabular:
6/1..3
3/1..2
2/1..1
pw
pw
pw
X
![Page 7: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/7.jpg)
7
Probability Distribution
The probability function – May be graphical:
1 2 3
.50
.33
.17
![Page 8: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/8.jpg)
8
Probability Distribution
The probability function– May be formulaic:
1,2,3for x6
4
xxXP
![Page 9: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/9.jpg)
9
Probability Distribution: Fair die
6/1..6
6/1..5
6/1..4
6/1..3
6/1..2
6/1..1
pw
pw
pw
pw
pw
pw
X
1 2 3
.50
.33
.17
4 5 6
![Page 10: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/10.jpg)
10
Probability Distribution
The probability function, properties
xxPX each for 0
x
X xP 1
![Page 11: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/11.jpg)
11
Cumulative Probability Distribution
Cumulative probability distribution– The cdf is a function which describes the probability
that a random variable does not exceed a value.
xXPxFX
Does this make sense for a continuous r.v.? Yes!
![Page 12: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/12.jpg)
12
Cumulative Probability Distribution
Cumulative probability distribution– The relationship between the cdf and the probability
function:
xy
XX yXPxXPxF
![Page 13: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/13.jpg)
13
Cumulative Probability Distribution
Die-throwing
66/6
656/5
546/4
436/3
326/2
216/1
10
x
x
x
x
x
x
x
xFX
1 2 3 4 5 6
1
graphicaltabular
xy
XX yXPxXPxF
( ) 1/ 6XP x P X x
![Page 14: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/14.jpg)
14
Cumulative Probability Distribution
The cumulative distribution function – May be formulaic (die-throwing):
min max x,0 ,6
6
floorP X x
![Page 15: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/15.jpg)
15
Cumulative Probability Distribution
The cdf, properties
xxFX each for 10
decreasing-non isxFX
right thefrom continuous isxFX
![Page 16: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/16.jpg)
16
Of a discrete probability distribution
Of a continuous probability distribution
Of a distribution which has both a continuous part and a discrete part.
Example CDFs
![Page 17: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/17.jpg)
17
Functions of a random variable
It is possible to calculate expectations and variances of functions of random variables
x
xXPxgXgE
x
xXPxgExgXgV 2
![Page 18: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/18.jpg)
18
Functions of a random variableExample
– You are paid a number of dollars equal to the square root of the number of spots on a die.
– What is a fair bet to get into this game?
x P(X=x) Product1 1 1/6 0.167
2 1.414 1/6 0.236
3 1.732 1/6 0.289
4 2 1/6 0.333
5 2.231 1/6 0.372
6 2.449 1/6 0.408
Tot 1.804
x
![Page 19: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/19.jpg)
19
Functions of a random variable
Linear functions– If a and b are constants and X is a random variable
– It can be shown that:
bXaEbaXE
XVabaXV 2Intuitively,why does b not appear in variance?And, why a2 ?
![Page 20: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/20.jpg)
20
The Most Common
Discrete Probability Distributions
(some discussed before)
1) --- Bernoulli distribution2) --- Binomial3) --- Geometric4) --- Poisson
![Page 21: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/21.jpg)
21
Bernoulli distribution
The Bernoulli distribution is the “coin flip” distribution.
X is Bernoulli if its probability function is:
1 . .
0 . . 1
w p pX
w p p
X=1 is usually interpreted as a “success.” E.g.:X=1 for heads in coin tossX=1 for male in surveyX=1 for defective in a test of productX=1 for “made the sale” tracking performance
![Page 22: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/22.jpg)
22
Bernoulli distribution
Expectation:
Variance:
pppXE 011
pppp
ppp
XEXEXV
1
0112
222
22
![Page 23: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/23.jpg)
23
Binomial distribution
The binomial distribution is just n independent Bernoullis
added up.
It is the number of “successes” in n trials.
If Z1, Z2, …, Zn are Bernoulli, then X is binomial:
nZZZX 21
![Page 24: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/24.jpg)
24
Binomial distribution
The binomial distribution is just n independent Bernoullis
added up. Testing for defects “with replacement.”
– Have many light bulbs
– Pick one at random, test for defect, put it back
– Pick one at random, test for defect, put it back
– If there are many light bulbs, do not have to replace
![Page 25: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/25.jpg)
Binomial distribution
Let’s figure out a binomial r.v.’s probability function.Suppose we are looking at a binomial with n=3.
We want P(X=0):– Can happen one way: 000– (1-p)(1-p)(1-p) = (1-p)3
We want P(X=1):– Can happen three ways: 100, 010, 001– p(1-p)(1-p)+(1-p)p(1-p)+(1-p)(1-p)p = 3p(1-p)2
We want P(X=2):– Can happen three ways: 110, 011, 101– pp(1-p)+(1-p)pp+p(1-p)p = 3p2(1-p)
We want P(X=3):– Can happen one way: 111– ppp = p3
![Page 26: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/26.jpg)
26
Binomial distribution
So, binomial r.v.’s probability function
3
2
2
3
0 . . 1
1 . . 3 1
2 . . 3 1
3 . .
w p p
w p p pX
w p p p
w p p
xnxX ppxP 1 waysof #
!1
! !n xxn
p px n x
![Page 27: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/27.jpg)
27
Binomial distribution
Typical shape of binomial:– Symmetric
![Page 28: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/28.jpg)
28
Expectation:
1 1 1
n n n
i ii i i
E X E Z E Z p np
Variance:
n
i
n
ii
n
ii
pnppp
ZVZVXV
1
11
11
Aside: ( ) ( ) 2 ( )V X Y V X V Y V X If V(X) = V(Y). And?
But (2 ) 4 ( )V X X V X V X Hmm…
![Page 29: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/29.jpg)
29
Binomial distribution
A salesman claims that he closes a deal 40% of the time.
This month, he closed 1 out of 10 deals.
How likely is it that he did 1/10 or worse given his claim?
![Page 30: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/30.jpg)
30
Binomial distribution
0 10
9
10!0 0.4 0.6 1 1 0.006 0.006
0! 10!
10!1 0.4 0.6 10 0.4 0.010 0.040
1! 9!
X
X
P
P
( 1) 0.046XP X
Less than 5% or1 in 20.So, it’s unlikelythat his successrate is 0.4.
4 6 10 9 8 710!
4 0.4 0.6 0.0256 0.0467 0.2514! 6! 4 3 2XP
Note:
![Page 31: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/31.jpg)
Binomial and normal / Gaussian distribution
The normal distributionis a good approximationto the binomial distribution. (“large” n,small skew.)
B(n, p)
Prob. density function:
![Page 32: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/32.jpg)
32
Geometric Distribution
A geometric distribution is usually interpreted as number of time periods until a failure occurs.
Imagine a sequence of coin flips, and the random variable X is the flip number on which the first tails occurs.
The probability of a head (a success) is p.
![Page 33: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/33.jpg)
Geometric
Let’s find the probability function for the geometric distribution:
pppppP
ppP
pP
X
X
X
113
12
11
2
ppxP xX 11
etc.
So, ?XP x (x is a positive integer)
![Page 34: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/34.jpg)
34
Geometric
Notice, there is no upper limit on how large X can be
Let’s check that these probabilities add to 1:
11
111
11
0
1
1
1
1
1
pppp
ppppxP
x
x
x
x
x
x
xX
Geometric series
![Page 35: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/35.jpg)
35
Geometric
Expectation:
pp
p
xpppxpxxPx
x
x
x
xX
1
1
1
11
11
2
1
1
1
1
1
21 p
pXV
Variance:
See Rosenpage 158,example 17.
0
1
1x
x
pp
(| | 1)p
differentiate both sides w.r.t. p:1
2 20
1 11 1
(1 ) (1 )x
x
xpp p
![Page 36: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/36.jpg)
36
Poisson distribution
The Poisson distribution is typical of random variables which represent counts.
– Number of requests to a server in 1 hour.
– Number of sick days in a year for an employee.
![Page 37: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/37.jpg)
37
The Poisson distribution is derived from the following underlying arrival time model:
– The probability of an unit arriving is uniform through time.
– Two items never arrive at exactly the same time.
– Arrivals are independent --- the arrival of one unit does not make the next unit more or less likely to arrive quickly.
![Page 38: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/38.jpg)
38
Poisson distribution
The probability function for the Poisson distribution with parameter is:
is like the arrival rate --- higher means more/faster arrivals
for x 0,1,2,3,...!
x
x
eP X x
x
XVXE
![Page 39: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/39.jpg)
39
Poisson distribution
Shape
Low Med High
![Page 40: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/40.jpg)
Markov and Chebyshev bounds
40
![Page 41: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/41.jpg)
Often, you don’t know the exact probability distribution
of a random variable.
We still would like to say something about the probabilities involving that random variable…
E.g., what is the probability of X being larger (or smaller) than some given value.
We often can by bounding the probability of events based on partial information about the underlying probability distribution
Markov and Chebyshev bounds.
![Page 42: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/42.jpg)
42
Theorem Markov Inequality
Let X be a nonnegative random variable with E[X] = .
Then, for any t > 0,( )P X t
t
Hmm. What if ? t ( ) 1P X t Sure!
Note: relatescumulative distributionto expected value.
2t gives 1
( 2 )2
P X But“Can’t have too muchprob. to the right of E[X]”
![Page 43: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/43.jpg)
43
Proof. ( ) [ ]t P X t E X
:
. ( ) . ( )x x t
t P X t t P X x
:
( )x x t
x P X x
( )x
x P X x [ ]E X
Where did we use X >= 0? 3rd line
t
( )P X x
x
[ ]( )
E XP X t
t I.e.
![Page 44: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/44.jpg)
44
Alt. proof Markov Inequality
( )P X tt
( ) 0I X
[ ]E X
Define0 X t
Yt X t
A discrete random variable
( ) 0( )
( )Y
P X t yp y
P X t y t
[ ] 0 ( ) ( )E Y P X t t P X t [ ]E X
0t
E[Y] E[X]
![Page 45: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/45.jpg)
Example:
Consider a system with mean time to failure = 100 hours.
Use the Markov inequality to bound the reliability of the system,
R(t) for t = 90, 100, 110, 200
By Markov
( 90) 100 / 90 1.11P X
( 100) 100 /100 1P X
( 110) 100 /110 0.9P X
( 200) 100 / 200 0.5P X
X – time to failure of the system; E[X]=100
R(t)= P[X>t] , with t =90, 100, 110 , 200
( )P X tt
Markov inequality is somewhat crude,
since only the mean is assumed to be known.
![Page 46: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/46.jpg)
46
Assume that mean and variance are given.
Better estimate of probability of events of interest using Chebyshev inequality:
Proof: Apply Markov inequality to non-negative r.v. (X- )2 and number t2 to obtain
Theorem Chebyshev's Inequality
![Page 47: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/47.jpg)
47
Theorem Chebyshev's Inequality
( )P X tt
0t
( ) 0I X [ ]E X
2
2
2
[ ]
[ ]
X
X
XX
E X
V X
P X tt
Alternate form
![Page 48: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/48.jpg)
48
Theorem Chebyshev's Inequality
( )P X tt
0t
( ) 0I X [ ]E X
2
2X
XP X tt
2 2X XP X t P X t
2
2
XE X
t
Because
![Page 49: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/49.jpg)
49
Chebyshev inequality: Alternate forms
Yet two other forms of Chebyshev’s ineqaulity:
2
2X
XP X tt
Says something about the probability ofbeing “k standard deviations from the mean”.
![Page 50: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/50.jpg)
50
XX XX XX
Theorem Chebyshev's Inequality
2
2X
XP X tt
2
1X XP X k
k
XX
2
11X XP X k
k
00.75
0.8890.934
X
![Page 51: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/51.jpg)
51
XX XX XX
Theorem Chebyshev's Inequality
2
2X
XP X tt
2
1X XP X k
k Facts:
XX
2
11X XP X k
k
00.75
0.8890.934
X
2~ ( , )X N
![Page 52: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/52.jpg)
52
Example
![Page 53: 1 Discrete Math CS 2800 Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d) 1) Probability Distributions 2) Markov and Chebyshev Bounds](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d445503460f94a20a2c/html5/thumbnails/53.jpg)
53
60
6
2
1X XP X k
k
84X 24 4X
4 1/16X XP X
1161000 62.5 70
( )P X tt
70( 84) 0.8
84P X
Aside “just” Markov: