Discrete Random Variables and Probability Distributions

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

6. Rolling 2 Dice – Probability Mass Function & CDF

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

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-1p

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)