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- 1. Outline TopicsProbability, sample space, random variable Probability distribution Expected value Variance MomentsLinear transformations of random variables Joint distributions Applied Statistics for Economics 2. Introduction to Probability Theory SFC - juliohuato@gmail.com Spring 2012 SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 2. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsTopicsProbability, sample space, random variableProbability distributionExpected valueVarianceMomentsLinear transformations of random variablesJoint distributions SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 3. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsTopics The main topics in this chapter are: random variables and probability distributions, expected values: mean and variance, and Two random variables jointly considered. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 4. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability The world in motion is viewed as a set of random processes or random experiments. Randomness means that, no matter how much our understanding of the world may advance, there is always an element of ignorance or uncertainty in such understanding. In other words: given speciﬁc causes, we don’t know fully which states of the world will result. Or, given speciﬁc states of the world, we don’t know fully what speciﬁcally caused such states of the world. In other words, we are uncertain or – more plainly said – ignorant about the speciﬁc causes or, alternatively, eﬀects involved in these processes. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 5. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability Examples of random processes: Your meeting the next person, SFC students commuting to school, residents of the U.S. producing new goods in a given year, etc. Why are they random? Because we are uncertain about the gender or the age of the next person you’ll meet, the commuting time of SFC students or the means of transportation they will use, the annual gross domestic product in the U.S. or its composition, etc. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 6. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability The mutually exclusive possible results of these experiments are called the outcomes. E.g. the next person you’ll meet could be female or male, young or old; SFC students may take a few or many minutes to commute to school; U.S. annual GDP may go up or down by some amount compared to the previous year, etc. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 7. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability Probability: the degree of belief that the outcome of an experiment will be a particular one. How to decide which probability to assign to a particular outcome of an experiment (e.g. that if you meet another person, the gender of such person will be female)? How to make this decision in a well-informed, disciplined, scientiﬁc way?1 One can only use experience – individual or collective – i.e. history. We may keep record of the gender of the people we meet over time and use the data compiled to inform our belief or look at records on the gender composition of the local population, etc. 1 In a sense, the whole purpose of statistics is to determine probabilities or, alternatively, expectations based on probabilities. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 8. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsSample space, event Sample space or population: the set of all the possible outcomes of the experiment. E.g. the sample space of the experiment of ﬂipping a coin once is: S = {H, T }.2 Event: a subset of the sample space, i.e. a set of one or more outcomes. E.g. the event (M ≤ 1) that your car will “need one repair at most” includes “no repairs” (M = 0) and “one repair” (M = 1). 2 We rule out ‘freak’ possibilities, like the coin landing on its edge. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 9. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsRandom variables Random variable (r.v.): a numerical summary of a random outcome. For example, G = g , where (e.g.) g is 0 if ‘male’ and 1 if ‘female’. The number of times your car needs repair during a given month: M = m, where m = 0, 1, 2, 3, . . .. The time it takes for SFC students to commute to school: T = t, where t is time in minutes. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 10. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsRandom variables There are discrete and continuous random variables. Gender, summarized as a 0 if ‘male’ and 1 if ‘female’, and the number of car repairs in a month are discrete random variables. The commuting time, if recorded in fractions of an hour – or even fractions of minutes and seconds, etc. can be regarded as a continuous r.v. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 11. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a discrete r.v. The probability distribution of a discrete r.v. is a list of all the values of the r.v. and the probabilities associated to each value of the r.v. By convention, the probabilities are a number between 0 and 1, where 0 means impossibility and 1 means full certainty; the probabilities over the sample space must add up to 1. E.g. let G = 0, 1 be the r.v. ‘gender of the next person you’ll meet’. Then: G Pr(G = g ) 0 0.45 1 0.55 SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 12. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a discrete r.v. Using the information in the probability distribution, you can compute the probability of a given event. E.g. the probability that you’ll meet ‘a male or a female’: Pr(G = 0 or G = 1) = Pr(G = 0)+Pr(G = 1) = 0.45+0.55 = 1 = 100%. In words, we are completely certain that you’ll meet either a male or a female the next time you meet a person. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 13. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a discrete r.v. Admittedly, the previous example is trivial. But consider the probability distribution of your car needing repair(s) in a given month. The r.v. ‘number of repairs’ needed is denoted as M: M Pr(M = m) 0 0.80 1 0.10 2 0.06 3 0.03 4 0.01 What’s the probability that the the car will need one or two repairs in a month? Answer: Pr(M = 1 or M = 2) = Pr(M = 1)+Pr(M = 2) = 0.10+0.06 = 0.16 = 16%. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 14. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a discrete r.v. The cumulative probability distribution (also known as a ‘cumulative distribution function’ or c.d.f.) is the probability that the random variable is less than or equal to a particular value. The ﬁrst two columns of the following table are the same as in the previous table. The last column gives the c.d.f.: M Pr(M = m) Pr(M ≤ m) 0 0.80 0.80 1 0.10 0.90 2 0.06 0.96 3 0.03 0.99 4 0.01 1.00 SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 15. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a discrete r.v. A binary discrete r.v. (e.g. G = 0, 1) is called a Bernoulli r.v. The Bernoulli distribution is: 1 with probability p G= 0 with probability 1 − p where p is the probability of the next person being ‘female’. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 16. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsProbability distribution of a continuous r.v. The cumulative probability distribution of a continuous r.v. is also the probability that the random variable is less than or equal to a particular value. The probability density function (p.d.f.) of a continuous random variable summarizes the probabilities for each value of the random variable. The mathematical description of the p.d.f. of a continuous variable requires that you’re familiar with calculus. So, we’ll skip it for now. NB: Strictly speaking, the probability that a continuous random variable has a particular value is zero. We can only speak of the probability of the random variable falling in a range (between two given values). NB2: The p.d.f. and the c.d.f. show the same information in diﬀerent formats. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 17. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsCharacteristics of a r.v. distribution In the practice of statistics, two basic measures are used extensively to characterize the distribution of a r.v.: the expected value or mean (or average) and the variance. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 18. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsExpected value The expected value of a r.v. X or E (X ) is the average value of the variable over many repeated trials. It is computed as a weighted average of the possible outcomes, where the weights are the probabilities of the outcomes. It is also called the mean of X and denoted by µX . For a discrete r.v.: k E (X ) = x1 p1 + x2 p2 + · · · + xk pk = x i pi i=1 E.g.: You loan $100 to your friend for a year at 10% interest. There’s a 99% chance he’ll repay the loan and 1% he won’t. What’s the expected value of your loan at maturity? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 19. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsExpected value E.g.: You loan $100 to your friend for a year at 10% interest. There’s a 99% chance he’ll repay the loan and 1% he won’t. What’s the expected value of your loan at maturity? Answer: ($110 × 0.99) + ($0 × .01) = $108.90 E.g.: What’s the expected value or average of the number of car repairs per month? See the table above. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 20. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsExpected value E.g.: What’s the expected value or average of the number of car repairs per month (M)? See the table above. Answer: E (M) = (0 × 0.80) + (1 × 0.10) + (2 × 0.06)+ (3 × 0.03) + (4 × 0.01) = 0.35 What does that mean? E.g.: In general, what’s the expected value of a Bernoulli r.v. with Pr(G = 1) = p and Pr(G = 0) = 1 − p? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 21. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsExpected value E.g.: In general, what’s the expected value of a Bernoulli r.v. with Pr(G = 1) = p and Pr(G = 0) = 1 − p? Answer: E (G ) = (1 × p) + (0 × (1 − p)) = p Note 1: Think of the operator E (.) as a function that transforms data on a variable by multiplying each value of the variable by its probability and then adding up all the products. Note 2: The formula for the expected value of a continuous r.v. requires calculus. So we’ll skip it for now. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 22. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsVariance and standard deviation The variance of a r.v. Y is: var(Y ) = σY = E [(Y − µY )2 ] 2 The standard deviation is the positive square root of the variance σY : 2 s.d.(Y ) = σY = + σY Basically, the s.d. gives the same information as the variance, but in units that are easier to understand. The units of the standard deviation are the same units as Y and µY . What is the intuition behind the variance and/or the standard deviation? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 23. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsVariance and standard deviation For a discrete r.v.: k var(Y ) = σY = E [(Y − µY )2 ] = 2 (yi − µY )2 pi i=1 k s.d.(Y ) = σY = (yi − µY )2 pi i=1 E.g.: What are the var. and s.d. of the number of car repairs per month (M)? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 24. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsVariance and standard deviation E.g.: What are the var. and s.d. of the number of car repairs per month (M)? Answer: var(M) = [(0−0.35)2 ×0.80]+[(1−0.35)2 ×0.10]+[(2−0.35)2 ×0.06] +[(3 − 0.35)2 × 0.03] + [(4 − 0.35)2 × 0.01] = 0.6475 √ s.d.(M) = 0.6475 ∼ 0.80 = E.g.: What are the var. and s.d. of a Bernoulli r.v.? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 25. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsVariance E.g.: What are the var. and s.d. of a Bernoulli r.v.? Answer: var(G ) = [(0 − p)2 × (1 − p)] + [(1 − p)2 × p] = p(1 − p) s.d.(G ) = p(1 − p) Note 1: Think of the operator var(.) as a function that transforms data on a variable by taking the distance or diﬀerence between each value of the variable and its mean, squaring that diﬀerence, multiplying it by the respective probability, and then adding up all the products. Note 1: Think of the operator s.d.(.) as a function that transforms data on a variable by taking the distance or diﬀerence between each value of the -variable and its mean, Statistics for Economicsdiﬀerence, to Probability SFC juliohuato@gmail.com Applied squaring that 2. Introduction
- 26. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMoments More formally, in statistics, the characteristics of the distribution of a r.v. are called moments. E (Y ) is the ﬁrst moment, E (Y 2 ) is the second moment, and E (Y r ) is the r th moment of Y . The ﬁrst moment is the mean and it is a measure of the center of the distribution, the second moment is a measure of its dispersion or spread, and r -th moments for r > 2 measure other aspects of the distribution’s shape. Clearly, the second moment of the distribution is intimately related to the variance. How? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 27. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMoments Two other measures of the shape (using higher moments) of a distribution are: Skewness: E [(Y − µY )3 ] Skewness = 3 σY For a symmetric distribution, the skewness is zero. If the distribution has a long left tail, the skewness is negative. If the distribution has a long right tail, the skewness is positive. Kurtosis: E [(Y − µY )4 ] Kurtosis = 4 σY For a distribution with heavy tails (outliers are likely), the kurtosis is to Probability SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction
- 28. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMean of a linear function of a r.v. Consider the income tax schedule: Y = 2, 000 + 0.8X where X is pre-tax earnings and Y is after-tax earnings. What is the marginal tax rate? Suppose an individual’s next year pre-tax earnings are a r.v. with mean 2 µX and variance σX . Since her pre-tax earnings are random, her after-tax earnings are random as well. With the following mean: E (Y ) = µY = 2, 000 + 0.8µX Why? Remember that the operator E (Y ) means “multiply each value of Y by its probability and add up the results.” SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 29. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsVariance of a linear function of a r.v. In turn, the variance of Y is: var(Y ) = σY = E [(Y − µY )2 ]. 2 Since Y = 2, 000 + 0.8X , then (Y − µY ) = (2, 000 + 0.8X ) − (2, 000 + 0.8µX ) = 0.8(X − µX ). Therefore: E (Y − µY )2 = E {[0.8(X − µX )2 ]} = 0.64E [(X − µX )2 ]. 2 2 That is: σY = 0.64σX . And taking the positive square root of that number: σY = 0.8σX SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 30. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMean and var. of a linear function of a r.v. More generally, if X and Y are r.v.’s related by Y = a + bX , then: µY = a + bµX 2 σY = b 2 σY 2 σY = bσY SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 31. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsTwo random variables We now deal with the distribution of two random variables considered together. The joint probability distribution of two random variables X and Y is the probability that the random variables take certain values at once or Pr (X = x, Y = y ). The marginal probability distribution of a random variable Y is its probability distribution in the context of its relationship with (an)other variable(s). SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 32. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMulti-variate distributions The following table shows relative frequencies (probabilities): Joint distribution of weather conditions and commuting times Rain (X = 0) No rain (X = 1) Total Long commute (Y = 0) 0.15 0.07 0.22 Short commute (Y = 1) 0.15 0.63 0.78 Total 0.30 0.70 1.00 The cells show the joint probabilities. The marginal probabilities (the marginal distribution) of Y can be calculated from the joint distribution of X and Y . If X can take l diﬀerent values x1 , . . . , xl , then: l Pr (Y = y ) = Pr (X = xi , Y = y ) i=1 SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 33. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsConditional distribution The conditional probability that Y takes the value y when X is known to take the value x is written Pr (Y = y |X = x). The conditional distribution of Y given X = x is: Pr (X = x, Y = y ) Pr (Y = y |X = x) = Pr (X = x) SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 34. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsConditional mean Consider the following table: Joint and conditional distribution of M and A M =0 M =1 M =2 M =3 M =4 Total Joint distribution Old car (A = 0) 0.35 0.065 0.05 0.025 0.01 0.50 New car (A = 1) 0.45 0.035 0.01 0.005 0.00 0.50 Total 0.8 0.1 0.06 0.03 0.01 1.00 Conditional distribution Pr(M | A = 0) 0.70 0.13 0.10 0.05 0.02 1.00 Pr(M | A = 1) 0.90 0.07 0.02 0.01 0.00 1.00 The conditional expectation of Y given X (or conditional mean of Y given X ) is the mean of the conditional distribution of Y given X. k E (Y |X = x) = yi Pr (Y = yi |X = x). i=1 SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 35. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsLaw of iterated expectations The mean height of adults is the weighted average of the mean height of men and the mean height of women, weighted by the proportions of men and women. More generally: l E (Y ) = E (Y |X = xi )Pr (X = xi ). i=1 In other terms: E (Y ) = E [E (Y |X )]. This is called the law of iterated expectations. If E (Y |X ) = 0 then E (Y ) = 0. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 36. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsConditional variance The variance of Y conditional on X is the variance of the conditional distribution of Y given X : k var(Y |X = x) = [yi − E (Y |X = x)]2 Pr (Y = yi |X = x). i=1 Example. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 37. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsIndependence Two r.v.’s X and Y are independently distributed (i.e. independent) if knowing the value of one of them gives no information about the other, that is, if the conditional distribution of Y given X equals the marginal distribution of Y . Formally, X and Y are independent if, for all values x and y , Pr(Y = y |X = x) = Pr(Y = y ) or Pr(X = x, Y = y ) = Pr(X = x) Pr(Y = y ) In other words, the joint distribution of X and Y is the product of their marginal distributions. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 38. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsCovariance The covariance between two r.v.’s X and Y measures the extent to which they move together. The covariance is the expected value of the product of the deviations of the variables from their expected values. The ﬁrst equation below is the general formula of the covariance. The second equation is speciﬁc to discrete r.v.’s and it assumes that X can take on l values and Y can take on k values: cov(X , Y ) = σXY = E [(X − µX )(Y − µY )] k l cov(X , Y ) = (xj − µX )(yi − µY ) Pr(X = xj , Y = yi ). i=1 j=1 Note that −∞ < σXY < +∞. How do you interpret the covariance formula? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 39. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsCorrelation The problem with the covariance is that it is not bounded. Its size depends on the units of X and Y and is, thus, hard to interpret. The correlation between X and Y is another measure of their covariation. But, unlike the covariance, the correlation eliminates the ‘units’ problem. Its formula is: cov(X , Y ) σXY corr(X , Y ) = = var(X ) var(Y ) σX σY Note that −1 ≤ corr(X , Y ) ≤ 1. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 40. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsCorrelation and conditional mean If E (Y |X = x) = E (Y ) = µY , then X and Y are uncorrelated. That is, cov(X , Y ) = 0 and cov(X , Y ) = 0. This follows from the law of iterated expectations. SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 41. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsMean and variance of sums of r.v.’s The mean of the sum of two r.v.’s, X and Y , is the sum of their means: E (X + Y ) = E (X ) + E (Y ) = µX + µY The variance of the sum of X and Y is the sum of their variance plus twice their covariance: 2 2 var(X + Y ) = var(X ) + var(Y ) + 2cov(X , Y ) = σX + σY + 2σXY If X and Y are independent, the covariance is zero and the variance of their sum is the sum of their variances: 2 2 var(X + Y ) = var(X ) + var(Y ) = σX + σY Why? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability
- 42. Outline Topics Probability, sample space, random variable Probability distribution Expected value Variance Moments Linear transformations of random variables Joint distributionsSums of r.v.’s Let X , Y , and V be r.v.’s and a, b, and c be constants. These facts follow from the deﬁnitions of mean, variance, covariance, and correlation: E (a + bX + cY ) = a + bµX + cµY var(a + bY ) = b 2 σY 2 var(aX + bY ) = a2 σX + 2abσXY + b 2 σY 2 2 E (Y 2 ) = σY + µ2 2 Y cov(a + bX + cV , Y ) E (XY ) = σXY + µX µY |σXY | ≤ 2 2 σX σY Can you prove them? SFC - juliohuato@gmail.com Applied Statistics for Economics 2. Introduction to Probability

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