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RANDOM VARIABLES, EXPECTATIONS, VARIANCES ETC.
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RANDOM VARIABLES, EXPECTATIONS, VARIANCES ETC.

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  1. RANDOM VARIABLES, EXPECTATIONS, VARIANCES ETC.

  2. Variable • Recall: • Variable: A characteristic of population or sample that is of interest for us. • Random variable: A function defined on the sample space S that associates a real number with each outcome in S.

  3. 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.)

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

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

  6. CUMULATIVE DENSITY FUNCTION (C.D.F.) • CDF of a r.v. X is defined as F(x)=P(X≤x). • Note that, P(a<X ≤b)=F(b)-F(a). • A function F(x) is a CDF for some r.v. X iff it satisfies F(x) is continuous from right F(x) is non-decreasing.

  7. Example • Consider tossing three fair coins. • Let X=number of heads observed. • S={TTT, TTH, THT, HTT, THH, HTH, HHT, HHH} • P(X=0)=P(X=3)=1/8; P(X=1)=P(X=2)=3/8

  8. Example • Let

  9. JOINT DISTRIBUTIONS • In many applications there are more than one random variables of interest, say X1, X2,…,Xk. JOINT DISCRETE DISTRIBUTIONS • The joint probability mass function (joint pmf) of the k-dimensional discrete rv X=(X1, X2,…,Xk) is

  10. 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

  11. Example • Tossing two fair dice  36 possible sample points • Let X: sum of the two dice; Y: |difference of the two dice| • For e.g.: • For (3,3), X=6 and Y=0. • For both (4,1) and (1,4), X=5, Y=3.

  12. Example • Joint pmf of (x,y) Empty cells are equal to 0. e.g. P(X=7,Y≤4)=f(7,0)+f(7,1)+f(7,2)+f(7,3)+f(7,4)=0+1/18+0+1/18+0=1/9

  13. 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

  14. Example • In the previous example,

  15. JOINT DISCRETE DISTRIBUTIONS • JOINT CDF: • F(x1,x2) is a cdf iff

  16. JOINT CONTINUOUS DISTRIBUTIONS • A k-dimensional vector valued rvX=(X1, X2,…,Xk) is said to be continuous if there is a function f(x1, x2,…, xk), called the joint probability density function (joint pdf), of X, such that the joint cdf can be given as

  17. JOINT CONTINUOUS DISTRIBUTIONS • A function f(x1, x2,…, xk) is the joint pdf 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

  18. JOINT CONTINUOUS DISTRIBUTIONS • If the pair (X1,X2) of discrete random variables has the joint pdf f(x1,x2), then the marginal pdfs of X1 and X2 are

  19. JOINT DISTRIBUTIONS • If X1, X2,…,Xk are independent from each other, then the joint pdf can be given as And the joint cdf can be written as

  20. 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,

  21. Example Statistical Analysis of Employment Discrimination Data (Example from Dudewicz & Mishra, 1988; data from Dawson, Hankey and Myers, 1982) Affected class might be a minority group or e.g. women

  22. Example, cont. • Does this data indicate discrimination against the affected class in promotions in this company? • Let X=(X1,X2,X3) where X1 is pay grade of an employee; X2 is an indicator of whether the employee is in the affected class or not; X3 is an indicator of whether the employee was promoted or not • x1={5,7,9,10,11,12,13,14}; x2={0,1}; x3={0,1}

  23. Example, cont. • E.g., in pay grade 10 of this occupation (X1=10) there were 102 members of the affected class and 695 members of the other classes. Seven percent of the affected class in pay grade 10 had been promoted, that is (102)(0.07)=7 individuals out of 102 had been promoted. • Out of 1950 employees, only 173 are in the affected class; this is not atypical in such studies.

  24. Example, cont. • E.g. probability of a randomly selected employee being in pay grade 10, being in the affected class, and promoted: P(X1=10,X2=1,X3=1)=7/1950=0.0036 (Probability function of a discrete 3 dimensional r.v.) • E.g. probability of a randomly selected employee being in pay grade 10 and promoted: P(X1=10, X3=1)= (7+56)/1950=0.0323 (Note: 8% of 695 -> 56) (marginal probability function of X1 and X3)

  25. Example, cont. • E.g. probability that an employee is in the other class (X2=0) given that the employee is in pay grade 10 (X1=10) and was promoted (X3=1): P(X2=0| X1=10, X3=1)= P(X1=10,X2=0,X3=1)/P(X1=10, X3=1) =(56/1950)/(63/1950)=0.89 (conditional probability) • probability that an employee is in the affected class (X2=1) given that the employee is in pay grade 10 (X1=10) and was promoted (X3=1): P(X2=1| X1=10, X3=1)=(7/1950)/(63/1950)=0.11

  26. Describing the Population • We’re interested in describing the population by computing various parameters. • For instance, we calculate the population mean and population variance.

  27. 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)]<.

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

  29. Population Mean (Expected Value) • Given a discrete random variable X with values xi, that occur with probabilities p(xi), the population mean of X is

  30. Population Variance • Let X be a discrete random variable with possible values xi that occur with probabilities p(xi), and let E(xi) =. The variance of X is defined by Unit*Unit Unit

  31. EXPECTED VALUE • The expected value or mean value of a continuous random variable X with pdf f(x) is • The variance of a continuous random • variable X with pdf f(x) is

  32. EXAMPLE • The pmf for the number of defective items in a lot is as follows Find the expected number and the variance of defective items. Results: E(X)=0.99, Var(X)=0.8699

  33. EXAMPLE • Let X be a random variable. Its pdf is f(x)=2(1-x), 0< x < 1 Find E(X) and Var(X).

  34. Laws of Expected Value • Let X be a rv and a, b, and c be constants. Then, for any two functions g1(x) and g2(x) whose expectations exist,

  35. 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) Laws of Expected Value and Variance Let X be a rv and c be a constant.

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

  37. 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

  38. EXPECTED VALUE If X1 and X2 are independent,

  39. CONDITIONAL EXPECTATION AND VARIANCE

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

  41. Example • An insect lays a large number of eggs, each surviving with probability p. Consider a large number of mothers. X: number of survivors in a litter; Y: number of eggs laid • Assume: • Find: expected number of survivors, i.e. E(X)

  42. Example - solution EX=E(E(X|Y)) =E(Yp) =p E(Y) =p E(E(Y|Λ)) =p E(Λ) =pβ

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

  44. SKEWNESS • Measure of lack of symmetry in the pdf. If the distribution of X is symmetric around its mean , 3=0  Skewness=0

  45. KURTOSIS • Measure of the peakedness of the pdf. Describes the shape of the distribution. Kurtosis=3  Normal Kurtosis >3  Leptokurtic (peaked and fat tails) Kurtosis<3  Platykurtic (less peaked and thinner tails)

  46. Measures of Central Location • Usually, we focus our attention on two types of measures when describing population characteristics: • Central location • Variability or spread

  47. With one data point clearly the central location is at the point itself. Measures of Central Location • The measure of central location reflects the locations of all the data points. • How? With two data points, the central location should fall in the middle between them (in order to reflect the location of both of them). But if the third data point appears on the left hand-side of the midrange, it should “pull” the central location to the left.

  48. Sum of the observations Number of observations Mean = The Arithmetic Mean • This is the most popular measure of central location

  49. The Arithmetic Mean Sample mean Population mean Sample size Population size

  50. The Arithmetic Mean • Example The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet. 0 7 22 11.0