1. STAT 552 PROBABILITY AND STATISTICS II
INTRODUCTION
Short review of S551

2. WHAT IS STATISTICS?
Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics is a way to get information from data. It is the science of uncertainty.

3. BASIC DEFINITIONS
POPULATION: The collection of all items of interest in a particular study.
SAMPLE: A set of data drawn from the population;
a subset of the population available for observation
PARAMETER: A descriptive measure of the
population, e.g., mean
STATISTIC: A descriptive measure of a sample
VARIABLE: A characteristic of interest about each
element of a population or sample.

4. STATISTIC
Statistic (or estimator) is any function of a r.v. of r.s. which do not contain any unknown quantity. E.g.
are statistics.
are NOT.
Any observed or particular value of an estimator is an estimate.

5. Sample Space
The set of all possible outcomes of an experiment is called a sample space and denoted by S.
Determining the outcomes.
Build an exhaustive list of all possible outcomes.
Make sure the listed outcomes are mutually exclusive.

6. RANDOM VARIABLES
Variables whose observed value is determined by chance
A r.v. is a function defined on the sample space S that associates a real number with each outcome in S.
Rvs are denoted by uppercase letters, and their observed values by lowercase letters.

7. DESCRIPTIVE STATISTICS
Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making.
Descriptive statistics methods make use of
graphical techniques
numerical descriptive measures.

8. Types of data – examples
Examples of types of data
Quantitative
Continuous
Discrete
Blood pressure, height, weight, age
Number of children
Number of attacks of asthma per week
Categorical (Qualitative)
Ordinal (Ordered categories)
Nominal (Unordered categories)
Grade of breast cancer
Better, same, worse
Disagree, neutral, agree
Sex (Male/female)
Alive or dead
Blood group O, A, B, AB

9. PROBABILITY
POPULATION
SAMPLE
STATISTICAL
INFERENCE

10. PROBABILITY: A numerical value expressing the degree of uncertainty regarding the occurrence of an event. A measure of uncertainty.
STATISTICAL INFERENCE: The science of drawing inferences about the population based only on a part of the population, sample.

11. Probability
P : S  [0,1] Probability domain range function

12. THE CALCULUS OF PROBABILITIES
If P is a probability function and A is any set, then
a. P()=0
b. P(A)  1
c. P(AC)=1  P(A)

13. ODDS The odds of an event A is defined by
It tells us how much more likely to see the
occurrence of event A.

14. ODDS RATIO OR is the ratio of two odds.
Useful for comparing the odds under two different conditions or for two different groups, e.g. odds for males versus females.

15. CONDITIONAL PROBABILITY
(Marginal) Probability: P(A): How likely is it that an event A will occur when an experiment is performed?
Conditional Probability: P(A|B): How will the probability of event A be affected by the knowledge of the occurrence or nonoccurrence of event B?
If two events are independent, then P(A|B)=P(A)

16. CONDITIONAL PROBABILITY

17. BAYES THEOREM
Suppose you have P(B|A), but need P(A|B).

18. Independence A and B are independent iff P(A|B)=P(A) or P(B|A)=P(B)
P(AB)=P(A)P(B)
A1, A2, …, An are mutually independent iff
for every subset j of {1,2,…,n}
E.g. for n=3, A1, A2, A3 are mutually independent iff P(A1A2A3)=P(A1)P(A2)P(A3) and P(A1A2)=P(A1)P(A2) and P(A1A3)=P(A1)P(A3) and P(A2A3)=P(A2)P(A3)

19. DISCRETE RANDOM VARIABLES
If the set of all possible values of a r.v. X is a countable set, then X is called discrete r.v.
The function f(x)=P(X=x) for x=x1,x2, … that assigns the probability to each value x is called probability density function (p.d.f.) or probability mass function (p.m.f.)

20. Example Discrete Uniform distribution:
Example: throw a fair die. P(X=1)=…=P(X=6)=1/6

21. CONTINUOUS RANDOM VARIABLES
When sample space is uncountable (continuous)
Example: Continuous Uniform(a,b)

22. CUMULATIVE DENSITY FUNCTION (C.D.F.)
CDF of a r.v. X is defined as F(x)=P(X≤x).

23. JOINT DISCRETE DISTRIBUTIONS
A function f(x1, x2,…, xk) is the joint pmf for some vector valued rv X=(X1, X2,…,Xk) iff the following properties are satisfied:
f(x1, x2,…, xk) 0 for all (x1, x2,…, xk)
and

24. MARGINAL DISCRETE DISTRIBUTIONS
If the pair (X1,X2) of discrete random variables has the joint pmf f(x1,x2), then the marginal pmfs of X1 and X2 are

25. CONDITIONAL DISTRIBUTIONS
If X1 and X2 are discrete or continuous random variables with joint pdf f(x1,x2), then the conditional pdf of X2 given X1=x1 is defined by
For independent rvs,

26. EXPECTED VALUES
Let X be a rv with pdf fX(x) and g(X) be a function of X. Then, the expected value (or the mean or the mathematical expectation) of g(X)
providing the sum or the integral exists, i.e.,
<E[g(X)]<.

27. EXPECTED VALUES
E[g(X)] is finite if E[| g(X) |] is finite.

28. Laws of Expected Value and Variance
Let X be a rv and c be a constant.
Laws of Expected Value
E(c) = c
E(X + c) = E(X) + c
E(cX) = cE(X)
Laws of Variance
V(c) = 0
V(X + c) = V(X)
V(cX) = c2V(X)

29. EXPECTED VALUE If X and Y are independent,
The covariance of X and Y is defined as

30. If (X,Y)~Normal, then X and Y are independent iff
EXPECTED VALUE
If X and Y are independent,
The reverse is usually not correct! It is only correct under normal distribution.
If (X,Y)~Normal, then X and Y are independent iff
Cov(X,Y)=0

31. EXPECTED VALUE
If X1 and X2 are independent,

32. CONDITIONAL EXPECTATION AND VARIANCE

33. CONDITIONAL EXPECTATION AND VARIANCE
(EVVE rule)
Proofs available in Casella & Berger (1990), pgs. 154 & 158

34. SOME MATHEMATICAL EXPECTATIONS
Population Mean:  = E(X)
Population Variance:
(measure of the deviation from the population mean)
Population Standard Deviation:
Moments:

35. The Variance
This measure reflects the dispersion of all the observations
The variance of a population of size N x1, x2,…,xN whose mean is m is defined as
The variance of a sample of n observations x1, x2, …,xn whose mean is is defined as

36. MOMENT GENERATING FUNCTION
The m.g.f. of random variable X is defined as
for t Є (-h,h) for some h>0.

37. Properties of m.g.f. M(0)=E[1]=1
If a r.v. X has m.g.f. M(t), then Y=aX+b has a m.g.f.
M.g.f does not always exists (e.g. Cauchy distribution)

38. CHARACTERISTIC FUNCTION
The c.h.f. of random variable X is defined as
for all real numbers t.
C.h.f. always exists.

39. Uniqueness
Theorem:
If two r.v.s have mg.f.s that exist and are equal, then they have the same distribution.
If two r.v.s have the same distribution, then they have the same m.g.f. (if they exist)
Similar statements are true for c.h.f.

40. SOME DISCRETE PROBABILITY DISTRIBUTIONS
Please review: Degenerate, Uniform, Bernoulli, Binomial, Poisson, Negative Binomial, Geometric, Hypergeometric, Extended Hypergeometric, Multinomial

41. SOME CONTINUOUS PROBABILITY DISTRIBUTIONS
Please review: Uniform, Normal (Gaussian), Exponential, Gamma, Chi-Square, Beta, Weibull, Cauchy, Log-Normal, t, F Distributions

42. TRANSFORMATION OF RANDOM VARIABLES
If X is an rv with pdf f(x), then Y=g(X) is also an rv. What is the pdf of Y?
If X is a discrete rv, replace Y=g(X) whereever you see X in the pdf of f(x) by using the relation
If X is a continuous rv, then do the same thing, but now multiply with Jacobian.
If it is not 1-to-1 transformation, divide the region into sub-regions for which we have 1-to-1 transformation.

43. CDF method
Example: Let
Consider . What is the p.d.f. of Y?
Solution:

44. M.G.F. Method
If X1,X2,…,Xn are independent random variables with MGFs Mxi (t), then the MGF of is

45. THE PROBABILITY INTEGRAL TRANSFORMATION
Let X have continuous cdf FX(x) and define the rv Y as Y=FX(x). Then,
Y ~ Uniform(0,1), that is,
P(Y  y) = y, 0<y<1.
This is very commonly used, especially in random number generation procedures.

46. SAMPLING DISTRIBUTION
A statistic is also a random variable. Its distribution depends on the distribution of the random sample and the form of the function Y=T(X1, X2,…,Xn). The probability distribution of a statistic Y is called the sampling distribution of Y.

47. SAMPLING FROM THE NORMAL DISTRIBUTION
Properties of the Sample Mean and Sample Variance
Let X1, X2,…,Xn be a r.s. of size n from a N(,2) distribution. Then,

48. SAMPLING FROM THE NORMAL DISTRIBUTION
If population variance is unknown, we use sample variance:

49. SAMPLING FROM THE NORMAL DISTRIBUTION
The F distribution allows us to compare the variances by giving the distribution of
If X~Fp,q, then 1/X~Fq,p.
If X~tq, then X2~F1,q.

50. CENTRAL LIMIT THEOREM Random Sample
If a random sample is drawn from any population, the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of will resemble a normal distribution.
Sample Mean Distribution X
Random Variable (Population) Distribution
Random Sample
(X1, X2, X3, …,Xn)

51. Sampling Distribution of the Sample Mean
If X is normal, is normal.
If X is non-normal, is approximately normally distributed for sample size greater than or equal to 30.