1. Review of Probability

2. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables 4 The Normal, Chi-Squared,, and t Distributions 5 Random Sampling and the Sampling Distribution 6 Large-Sample Approximations to Sampling Distributions

3. Definitions  Outcomes: the mutually exclusive potential results of a random process.  Probability: the proportion of the time that the outcome occurs.  Sample space: the set of all possible outcomes.  Event: A subset of the sample space.  Random variables: a random variable is a numerical summary of a random outcome.

4. Probability distribution: discrete variable  List of all possible [x, p(x)] pairs x = value of random variable (outcome) p(x) = probability associated with value  Mutually exclusive (no overlap)  Collectively exhaustive (nothing left out)  0  p(x)  1 for all x   p(x) = 1

5. Probability distribution: discrete variable  Probabilities of events.  Cumulative probability distribution.

7. Example: Bernoulli distribution.  Let G be the gender of the next new person you meet, where G=0 indicates that the person is male and G=1 indicates that she is female.  The outcomes of G and their probabilities are G= 1 with probability p = 0 with probability 1-p

8. Probability distribution: continuous variable

9. 1. Mathematical formula 2. Shows all values, x, and frequencies, f(x) f(x) Is Not Probability Value (Value, Frequency) Frequency f(x) ab x (Area Under Curve) fxdx fx () () All x a x b    1 0, 3.Properties

10.  Probability density function (p.d.f.).

11. Probability distribution: Continuous Variable  Cumulative probability distribution.

12. x f(x)f(x) dcab Uniform Distribution 3. Mean and Standard Deviation 2. Probability density function 1. Equally likely outcomes

13. Expected Values, Mean, and Variance

14. Expected value of a Bernoulli random variable

15. Expected value of a continuous random variable  Let f(Y) is the p.d.f of random variable Y, then the expected value of Y is

16. Variance, Standard Deviation, and Moments

17. Variance of a Bernoulli random variable The mean of the Bernoulli random variable G is, so its variance is The standard deviation is

18. Moments  The expected value of Y r is called the r th moments of the random variable Y. That is the r th moment of Y is E(Y r ).  The mean of Y, E(Y), is also called the first moment of Y.

19. Moments, ctd.  skewness = =measure of asymmetry of a distribution  skewness = 0: distribution is symmetric  skewness > (<) 0: distribution has long right (left) tail

20. Moments, ctd.  kurtosis = =measure of mass in tails = measure of probability of large values  kurtosis = 3: normal distribution  kurtosis > 3: heavy tails (“leptokurtotic”)

22. Mean and Variance of a Linear Function of a Random Variable Suppose X is a random variable with mean and variance, and Then the mean and variance of Y are and the standard deviation of Y is

23. Two Random Variables Joint and Marginal Distributions  The joint probability distribution of two discrete random variables, say X and Y, is the probability that the random variables simultaneously take on certain values, say x and.  The joint probability distribution can be written as the function

25.  The marginal probability distribution of a random variable Y is just another name for its probability distribution.

27. Conditional distribution of Y given X=x is Conditional expectation of Y given X=x is

30.  The mean of Y is the weighted average of the conditional expectation of Y given X, weighted by the probability distribution of X.

31.  Stated differently, the expectation of Y is the expectation of the conditional expectation of Y given X, that is where the inner expectation is computed using the conditional distribution of Y given X and the outer expectation is computed using the marginal distribution of X.  This is known as the law of iterated expectations.

33. Conditional variance  The variance of Y conditional on X is the variance of the conditional distribution of Y given X.

35. Independence  Two random variable X and Y are independently distributed, or independent, if knowing the value of one of the variables provides no information about the other.  That is, X and Y are independent if for all values of x and,

36.  State differently, X and Y are independent if  That is, the joint distribution of two independent random variables is the product of their marginal distributions.

37. Covariance and Correlation Covariance One measure of the extent to which two random variables move together is their covariance.

38. Correlation  The correlation is an alternative measure of dependence between X and Y that solves the “unit” problem of covariance.  The random variables X and Y are said to be uncorrelated if Corr(X, Y) = 0.  The correlation is always between -1 and 1.

40. The Mean and Variance of Sums of Random Variables

42. Normal, Chi-Squared,, and t Distributions  The Normal Distribution The probability density function of a normal distributed random variable (the normal p.d.f.) is where exp(x) is the exponential function of x. The factor ensures that

43. The normal distribution with mean μ and variance σ 2 is expressed as N(μ, σ 2 ).

44.  90%: +- 1.69  95%: +- 1.96  99%: +- 2.58

45. The Empirical Rule (normal distribution) Copyright © 2011 Pearson Education, Inc. 36 of 42

46.  The standard normal distribution is the normal distribution with mean μ= 0 and variance σ 2 = 1 and is denoted N(0, 1).  The standard normal distribution is often denoted by Z and its cumulative distribution function is denoted by Ф. Accordingly, Pr(Z ≤ c)= Ф(c), where c is a constant.

47. Copyright © 2003 by Pearson Education, Inc. 2-47 Key Concept 2.4

49. The bivariate normal distribution  The bivariate normal p.d.f. for the two random variables X and Y is where is the correlation between X and Y.

50. Important properties for normal distribution. 1. If X and Y have a bivariate normal distribution with covariance, and if a and b are two constants, then 2. The marginal distribution of each of the two variables is normal. This follows by setting a = 1; b = 0 in 1. 3. If = 0, then X and Y are independent. 4. Any linear combination of random draws from normal distributions also has a normal distribution.

51. The Chi-squared distribution  The Chi-squared distribution is the distribution of the sum of m squared independent standard normal random variables.  The distribution depends on m, which is called the degrees of freedom of the chi-squared distribution.  A chi-squared distribution with m degrees of freedom is denoted.

52. distribution  where and are independent.  When n is ∞,.  The distribution is the distribution of a random variable with a chi-squared distribution with m degrees of freedom, divided by m.  Equivalently, the distribution is the distribution of the average of m squared standard normal random variables.

53. The Student t Distribution  The Student t distribution with m degrees of freedom is defined to be the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom divided by m.

54.  That is, let Z be a standard normal random variable, let W be a random variable with a chi-squared distribution with m degrees of freedom, and let Z and W be independently distributed. Then  When m is 30 or more, the Student t distribution is well approximated by the standard normal distribution, and t ∞ distribution equals the standard normal distribution Z.

55. Random Sampling  Simple random sampling is the simplest sampling scheme in which n objects are selected at random from a population and each member of the population is equally likely to be included in the sample.  Since the members of the population included in the sample are selected at random, the values of the observations Y 1, …, Y n are themselves random.

56. i.i.d. draws.  Because individuals #1 and #2 are selected at random, the value of Y 1 has no information content for Y 2. Thus: Y 1 and Y 2 are independently distributed Y 1 and Y 2 come from the same distribution, that is, Y 1, Y 2 are identically distributed That is, under simple random sampling, Y 1 and Y 2 are independently and identically distributed (i.i.d.). More generally, under simple random sampling, {Y i }, i = 1,…, n, are i.i.d.  This framework allows rigorous statistical inferences about moments of population distributions using a sample of data from that population …

57. Sampling Distribution of the Sample Average The sample average of the n observations Y 1, …, Y n is  Because Y 1, …, Y n are random, their average is random and has a probability distribution. The distribution of is called the sampling distribution of.

58. Mean and Variance of Suppose Y 1, …, Y n are i.i.d. and let and denote the mean and variance of Y i. Then

59. Things we want to know about the sampling distribution:  What is the mean of ? If E( ) = true  =.78, then is an unbiased estimator of   What is the variance of ? How does var( ) depend on n (famous 1/n formula)  Does become close to  when n is large? Law of large numbers: is a consistent estimator of   –  appears bell shaped for n large…is this generally true? In fact, –  is approximately normally distributed for n large (Central Limit Theorem)

60.  Find out the sampling distribution of if Y is normally distributed  The linear combination of normally distributed random variable is also normally distributed (equation 2. 42)  For a normal, we need to find out mean and variance to determine its distribution  Therefore,

61. Large-Sample Approximations to Sampling Distributions Two approaches to characterizing sample distributions.  Exact distribution, or finite sample distribution when the distribution of Y is known.  Asymptotic distribution: large-sample approximation to the sampling distribution.

62. Law of Large Numbers  The law of large numbers states that, under general conditions, will be near with very high probability when n is large.  The property that is near with increasing probability as n increases is called convergence in probability, or consistency.  The law of large numbers states that, under certain conditions, converges in probability to, or, is consistent for.

63. Copyright © 2003 by Pearson Education, Inc. 2-63 Key Concept 2.6

64. The conditions for the law of large numbers are  Y i, i=1, …, n, are i.i.d.  The variance of Y i,, is finite.

66. Formal definitions of consistency and law of large numbers. Consistency and convergence in probability.  Let S 1, S 2, …, S n, … be a sequence of random variables. For example, S n could be the sample average of a sample of n observations of the random variable Y.  The sequence of random variables {S n } is said to converge in probability to a limit, μ, if the probability that S n is within ±δ of μ tends to one as n → ∞, as long as the constant is positive.

67.  That is, if and only if Pr [| S n – μ |≥δ] → 0 as n → ∞ for every δ > 0.  If, then S n is said to be a consistent estimator of μ.  The law of large numbers. If Y 1, …, Y n are i.i.d., E(Y i ) = and Var(Y i ) < ∞, then

68. The Central Limit Theorem  The central limit theorem says that, under general conditions, the distribution of is well approximated by a normal distribution when n is large.  Since the mean of is and its variance if, when n is large the distribution of is approximately N(, ).  Accordingly, is well approximated by the standard normal distribution N(0,1)

69. The sampling distribution of when n is large  For small sample sizes, the distribution of is complicated, but if n is large, the sampling distribution is simple!  As n increases, the distribution of becomes more tightly centered around  Y (the Law of Large Numbers)  Moreover, the distribution of –  Y becomes normal (the Central Limit Theorem)

72. Convergence in distribution.  Let F 1, …, F n, … be a sequence of cumulative distribution functions corresponding to a sequence of random variables, S 1, …, S n, ….  Then the sequence of random variables S n is said to converge in distribution to S (denoted as ) if the distribution functions {F n } converge to F.

73.  That is, if and only if where the limit holds at all points t at which the limiting distribution F is continuous.  The distribution F is called the asymptotic distribution of S n.

74. The central limit theorem If Y 1, …, Y n are i.i.d. and 0 < < ∞, then In other words, the asymptotic distribution of is N(0,1).

75. Slutsky’s theorem  Slutsky’s theorem combines consistency and convergence in distribution.  Suppose that, where a is a constant, and. Then

76. Continuous mapping theorem If g is a continuous function, then 

77. But, how large of n is “large enough?”  The answer is: it depends on the distribution of the underlying Y i that make up the average.  At one extreme, if the Y i are themselves normally distributed, then is exactly normally distributed for all n.  In contrast, when Y i is far from normally distributed, then this approximation can require n = 30 or even more.

78. Example: A skewed distribution.