1. Prof. Bart Selman selman@cs.cornell.edu Module Probability --- Part d)
Discrete Math CS 2800
Prof. Bart Selman
Module
Probability --- Part d)
1) Probability Distributions
2) Markov and Chebyshev Bounds

2. 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

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 = 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

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.

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):

6. Probability Distribution
The probability function
May be tabular:

7. Probability Distribution
The probability function
May be graphical:
.50
.33
.17 1 2 3

8. Probability Distribution
The probability function
May be formulaic:

9. Probability Distribution: Fair die1 2 3
.50
.33
.17 4 5 6

10. Probability Distribution
The probability function, properties

11. Cumulative Probability Distribution
The cdf is a function which describes the probability that a random variable does not exceed a value.
Yes!
Does this make sense for a continuous r.v.?

12. Cumulative Probability Distribution
The relationship between the cdf and the probability function:

13. Cumulative Probability Distribution
Die-throwing
tabular
graphical 1 2 3 4 5 6

14. Cumulative Probability Distribution
The cumulative distribution function
May be formulaic (die-throwing):

15. Cumulative Probability Distribution
The cdf, properties

16. Example CDFs Of a discrete probability distribution
Of a continuous probability distribution
Of a distribution which has both a continuous part and a discrete part.

17. Functions of a random variable
It is possible to calculate expectations and variances of functions of random variables

18. Functions of a random variable
Example
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)
Product 1
1/6
0.167 2
1.414
0.236 3
1.732
0.289 4
0.333 5
2.231
0.372 6
2.449
0.408
Tot
1.804

19. Functions of a random variable
Functions of a random variable
Linear functions
If a and b are constants and X is a random variable
It can be shown that:
Intuitively,
why does b not appear in variance?
And, why a2 ?

20. Discrete Probability Distributions (some discussed before)
The Most Common
Discrete Probability Distributions
(some discussed before)
1) --- Bernoulli distribution
2) --- Binomial
3) --- Geometric
4) --- Poisson

21. Bernoulli distribution
The Bernoulli distribution is the “coin flip” distribution.
X is Bernoulli if its probability function is:
X=1 is usually interpreted as a “success.” E.g.:
X=1 for heads in coin toss
X=1 for male in survey
X=1 for defective in a test of product
X=1 for “made the sale” tracking performance

22. Bernoulli distribution
Expectation:
Variance:

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:

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
If there are many light bulbs, do not have to replace

25. 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

26. Binomial distribution
So, binomial r.v.’s probability function

27. Binomial distribution
Typical shape of binomial:
Symmetric

28. Expectation:
Variance:
Aside:
If V(X) = V(Y). And?
But
Hmm…

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?

30. Binomial distribution
Less than 5% or
1 in 20.
So, it’s unlikely
that his success
rate is 0.4.
Note:

31. Binomial and normal / Gaussian distribution
The normal distribution
is a good approximation
to the binomial
distribution. (“large” n,
small skew.)
B(n, p)
Prob. density function:

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.

33. Geometric
Let’s find the probability function for the geometric distribution:
etc.
So,
(x is a positive integer)

34. Geometric Geometric series
Notice, there is no upper limit on how large X can be
Let’s check that these probabilities add to 1:
Geometric series

35. Geometric differentiate both sides w.r.t. p: See Rosen page 158,
example 17.
Expectation:
Variance:

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.

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.

38. Poisson distribution
The probability function for the Poisson distribution with parameter  is:
 is like the arrival rate --- higher means more/faster arrivals

39. Poisson distribution
Shape
Low 
Med 
High 

40. Markov and Chebyshev bounds

41. Markov and Chebyshev bounds.
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.

42. Theorem  Markov Inequality
Note: relates
cumulative distribution
to expected value.
Theorem  Markov Inequality
Let X be a nonnegative random variable with E[X] = .
Then, for any t > 0,
Hmm. What if ?
Sure! 
“Can’t have too much
prob. to the right of E[X]”
But
gives

43. Proof
I.e.
Where did we use X >= 0?
3rd line

44. Alt. proof  Markov Inequality
Define
A discrete random variable
 E[Y] E[X]

45. X – time to failure of the system; E[X]=100
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
X – time to failure of the system; E[X]=100
R(t)= P[X>t] , with t =90, 100, 110 , 200
By Markov 
Markov inequality is somewhat crude,
since only the mean is assumed to be known.

46. Theorem  Chebyshev's Inequality
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

47. Theorem  Chebyshev's Inequality
Alternate form

48. Theorem  Chebyshev's Inequality
Because

49. Chebyshev inequality: Alternate forms
Yet two other forms of Chebyshev’s ineqaulity:
Says something about the probability of
being “k standard deviations from the mean”.

50. Theorem  Chebyshev's Inequality

51. Theorem  Chebyshev's Inequality
Facts:

52. Example

53. Aside “just” Markov: