1. Discrete Random Variables and Probability Distributions

2. Random Variables
Random Variable (RV): A numeric outcome that results from an experiment
For each element of an experiment’s sample space, the random variable can take on exactly one value
Discrete Random Variable: An RV that can take on only a finite or countably infinite set of outcomes
Continuous Random Variable: An RV that can take on any value along a continuum (but may be reported “discretely”
Random Variables are denoted by upper case letters (Y)
Individual outcomes for RV are denoted by lower case letters (y)

3. Probability Distributions
Probability Distribution: Table, Graph, or Formula that describes values a random variable can take on, and its corresponding probability (discrete RV) or density (continuous RV)
Discrete Probability Distribution: Assigns probabilities (masses) to the individual outcomes
Continuous Probability Distribution: Assigns density at individual points, probability of ranges can be obtained by integrating density function
Discrete Probabilities denoted by: p(y) = P(Y=y)
Continuous Densities denoted by: f(y)
Cumulative Distribution Function: F(y) = P(Y≤y)

4. Discrete Probability Distributions

5. Example – Rolling 2 Dice (Red/Green)
Y = Sum of the up faces of the two die. Table gives value of y for all elements in S
Red\Green 1 2 3 4 5 6 7 8 9 10 11 12

6. Rolling 2 Dice – Probability Mass Function & CDF
p(y)
F(y) 2
1/36 3
2/36
3/36 4
6/36 5
4/36
10/36 6
5/36
15/36 7
21/36 8
26/36 9
30/36 10
33/36 11
35/36 12
36/36

7. Rolling 2 Dice – Probability Mass Function

8. Rolling 2 Dice – Cumulative Distribution Function

9. Expected Values of Discrete RV’s
Mean (aka Expected Value) – Long-Run average value an RV (or function of RV) will take on
Variance – Average squared deviation between a realization of an RV (or function of RV) and its mean
Standard Deviation – Positive Square Root of Variance (in same units as the data)
Notation:
Mean: E(Y) = m
Variance: V(Y) = s2
Standard Deviation: s

10. Expected Values of Discrete RV’s

11. Expected Values of Linear Functions of Discrete RV’s

12. Example – Rolling 2 Dice y p(y) yp(y) y2p(y) 2 1/36 2/36 4/36 3 6/36
18/36 4
3/36
12/36
48/36 5
20/36
100/36 6
5/36
30/36
180/36 7
42/36
294/36 8
40/36
320/36 9
36/36
324/36 10
300/36 11
22/36
242/36 12
144/36
Sum
36/36=1.00
252/36=7.00
1974/36=54.833

13. Tchebysheff’s Theorem/Empirical Rule
Tchebysheff: Suppose Y is any random variable with mean m and standard deviation s. Then: P(m-ks ≤ Y ≤ m+ks) ≥ 1-(1/k2) for k ≥ 1
k=1: P(m-1s ≤ Y ≤ m+1s) ≥ 1-(1/12) = 0 (trivial result)
k=2: P(m-2s ≤ Y ≤ m+2s) ≥ 1-(1/22) = ¾
k=3: P(m-3s ≤ Y ≤ m+3s) ≥ 1-(1/32) = 8/9
Note that this is a very conservative bound, but that it works for any distribution
Empirical Rule (Mound Shaped Distributions)
k=1: P(m-1s ≤ Y ≤ m+1s)  0.68
k=2: P(m-2s ≤ Y ≤ m+2s)  0.95
k=3: P(m-3s ≤ Y ≤ m+3s)  1

14. Proof of Tchebysheff’s Theorem

15. Moment Generating Functions (I)

16. Moment Generating Functions (II)
M(t) is called the moment-generating function for Y, and cam be used to derive any non-central moments of the random variable (assuming it exists in a neighborhood around t=0).
Also, useful in determining the distributions of functions of rndom variables

17. Probability Generating Functions
P(t) is the probability generating function for Y

18. Discrete Uniform Distribution
Suppose Y can take on any integer value between a and b inclusive, each equally likely (e.g. rolling a dice, where a=1 and b=6). Then Y follows the discrete uniform distribution.

19. Bernoulli Distribution
An experiment consists of one trial. It can result in one of 2 outcomes: Success or Failure (or a characteristic being Present or Absent).
Probability of Success is p (0<p<1)
Y = 1 if Success (Characteristic Present), 0 if not

20. Binomial Experiment
Experiment consists of a series of n identical trials
Each trial can end in one of 2 outcomes: Success or Failure
Trials are independent (outcome of one has no bearing on outcomes of others)
Probability of Success, p, is constant for all trials
Random Variable Y, is the number of Successes in the n trials is said to follow Binomial Distribution with parameters n and p
Y can take on the values y=0,1,…,n
Notation: Y~Bin(n,p)

21. Binomial Distribution

25. Binomial Distribution – Expected Value

26. Binomial Distribution – Variance and S.D.

27. Binomial Distribution – MGF & PGF

28. Geometric Distribution
Used to model the number of Bernoulli trials needed until the first Success occurs (P(S)=p)
First Success on Trial 1  S, y = 1  p(1)=p
First Success on Trial 2  FS, y = 2  p(2)=(1-p)p
First Success on Trial k  F…FS, y = k  p(k)=(1-p)k-1 p

29. Geometric Distribution - Expectations

30. Geometric Distribution – MGF & PGF

31. Negative Binomial Distribution
Used to model the number of trials needed until the rth Success (extension of Geometric distribution)
Based on there being r-1 Successes in first y-1 trials, followed by a Success

32. Poisson Distribution
Distribution often used to model the number of incidences of some characteristic in time or space:
Arrivals of customers in a queue
Numbers of flaws in a roll of fabric
Number of typos per page of text.
Distribution obtained as follows:
Break down the “area” into many small “pieces” (n pieces)
Each “piece” can have only 0 or 1 occurrences (p=P(1))
Let l=np ≡ Average number of occurrences over “area”
Y ≡ # occurrences in “area” is sum of 0s & 1s over “pieces”
Y ~ Bin(n,p) with p = l/n
Take limit of Binomial Distribution as n  with p = l/n

33. Poisson Distribution - Derivation

34. Poisson Distribution - Expectations

35. Poisson Distribution – MGF & PGF

36. Hypergeometric Distribution
Finite population generalization of Binomial Distribution
Population:
N Elements
k Successes (elements with characteristic if interest)
Sample:
n Elements
Y = # of Successes in sample (y = 0,1,,,,,min(n,k)