Exam 2: Review of Probability

Two variables are uncorrelated in all of the cases below,with the exception of

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables.That is,show that corr(X,Y)= corr(X*,Y*),where X* = a + bX and Y* = c + dY,and where a,b,c,and d are arbitrary non-zero constants.

Use the definition for the conditional distribution of Y given X = x and the marginal distribution of X to derive the formula for Pr(X = x,Y = y).This is called the multiplication rule.Use it to derive the probability for drawing two aces randomly from a deck of cards (no joker),where you do not replace the card after the first draw.Next,generalizing the multiplication rule and assuming independence,find the probability of having four girls in a family with four children.

You have read about the so-called catch-up theory by economic historians,whereby nations that are further behind in per capita income grow faster subsequently.If this is true systematically,then eventually laggards will reach the leader.To put the theory to the test,you collect data on relative (to the United States)per capita income for two years,1960 and 1990,for 24 OECD countries.You think of these countries as a population you want to describe,rather than a sample from which you want to infer behavior of a larger population.The relevant data for this question is as follows: where X1 and X2 are per capita income relative to the United States in 1960 and 1990 respectively,and Y is the average annual growth rate in X over the 1960-1990 period.Numbers in the last row represent sums of the columns above. (a)Calculate the variance and standard deviation of X1 and X2.For a catch-up effect to be present,what relationship must the two standard deviations show? Is this the case here? (b)Calculate the correlation between Y and .What sign must the correlation coefficient have for there to be evidence of a catch-up effect? Explain.

A few years ago the news magazine The Economist listed some of the stranger explanations used in the past to predict presidential election outcomes.These included whether or not the hemlines of women's skirts went up or down,stock market performances,baseball World Series wins by an American League team,etc.Thinking about this problem more seriously,you decide to analyze whether or not the presidential candidate for a certain party did better if his party controlled the house.Accordingly you collect data for the last 34 presidential elections.You think of this data as comprising a population which you want to describe,rather than a sample from which you want to infer behavior of a larger population.You generate the accompanying table: Joint Distribution of Presidential Party Affiliation and Party Control of House of Representatives,1860-1996 (a)Interpret one of the joint probabilities and one of the marginal probabilities. (b)Compute E(X).How does this differ from E(X = 0)? Explain. (c)If you picked one of the Republican presidents at random,what is the probability that during his term the Democrats had control of the House? (d)What would the joint distribution look like under independence? Check your results by calculating the two conditional distributions and compare these to the marginal distribution.

The accompanying table lists the outcomes and the cumulative probability distribution for a student renting videos during the week while on campus. Video Rentals per Week during Semester Sketch the probability distribution.Next,calculate the cumulative probability distribution for the above table.What is the probability of the student renting between 2 and 4 a week? Of less than 3 a week?

Two random variables are independently distributed if their joint distribution is the product of their marginal distributions.It is intuitively easier to understand that two random variables are independently distributed if all conditional distributions of Y given X are equal.Derive one of the two conditions from the other.

The following problem is frequently encountered in the case of a rare disease,say AIDS,when determining the probability of actually having the disease after testing positively for HIV.(This is often known as the accuracy of the test given that you have the disease. )Let us set up the problem as follows: Y = 0 if you tested negative using the ELISA test for HIV,Y = 1 if you tested positive;X = 1 if you have HIV,X = 0 if you do not have HIV.Assume that 0.1 percent of the population has HIV and that the accuracy of the test is 0.95 in both cases of (i)testing positive when you have HIV,and (ii)testing negative when you do not have HIV.(The actual ELISA test is actually 99.7 percent accurate when you have HIV,and 98.5 percent accurate when you do not have HIV. ) (a)Assuming arbitrarily a population of 10,000,000 people,use the accompanying table to first enter the column totals. (b)Use the conditional probabilities to fill in the joint absolute frequencies. (c)Fill in the marginal absolute frequencies for testing positive and negative.Determine the conditional probability of having HIV when you have tested positive.Explain this surprising result. (d)The previous problem is an application of Bayes' theorem,which converts Pr(Y = y = x)into Pr(X = x = y).Can you think of other examples where Pr(Y = y = x)≠ Pr(X = x = y)?

The textbook mentioned that the mean of Y,E(Y)is called the first moment of Y,and that the expected value of the square of Y,E(Y2)is called the second moment of Y,and so on.These are also referred to as moments about the origin.A related concept is moments about the mean,which are defined as E[(Y -µY)r].What do you call the second moment about the mean? What do you think the third moment,referred to as "skewness," measures? Do you believe that it would be positive or negative for an earnings distribution? What measure of the third moment around the mean do you get for a normal distribution?

When there are ∞ degrees of freedom,the t∞ distribution

Think of an example involving five possible quantitative outcomes of a discrete random variable and attach a probability to each one of these outcomes.Display the outcomes,probability distribution,and cumulative probability distribution in a table.Sketch both the probability distribution and the cumulative probability distribution.