Exam 2: Review of Probability

For a normal distribution, the skewness and kurtosis measures are as follows:

The skewness is most likely positive for one of the following distributions:

The skewness of the distribution of a random variable Y is defined as follows:

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: Y \times 0.023 0.770 1.030 0.018 0.00053 0.593 1.0609 0.014 1.000 1.000 0.014 .00020 1.000 1.0000 \ldots 0.041 0.200 0.450 0.008 0.00168 0.040 0.2025 0.033 0.130 0.230 .004 0.00109 0.017 0.0529 0.625 13.220 17.800 0.294 0.01877 8.529 13.9164 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.

The height of male students at your college/university is normally distributed with a mean of 70 inches and a standard deviation of 3.5 inches. If you had a list of telephone numbers for male students for the purpose of conducting a survey, what would be the probability of randomly calling one of these students whose height is (a)taller than 6'0"? (b)between 5'3" and 6'5"? (c)shorter than 5'7", the mean height of female students? (d)shorter than 5'0"? (e)taller than Shaquille O'Neal, the center of the Boston Celtics, who is 7'1" tall? Compare this to the probability of a woman being pregnant for 10 months (300 days), where days of pregnancy is normally distributed with a mean of 266 days and a standard deviation of 16 days.

The accompanying table gives the outcomes and probability distribution of the number of times a student checks her e-mail daily: Probability of Checking E-Mail Outcome number of e- mail checks) 0 1 2 3 4 5 6 Probability distribution 0.05 0.15 0.30 0.25 0.15 0.08 0.02 Sketch the probability distribution. Next, calculate the c.d.f. for the above table. What is the probability of her checking her e-mail between 1 and 3 times a day? Of checking it more than 3 times a day?

Probabilities and relative frequencies are related in that the probability of an outcome is the proportion of the time that the outcome occurs in the long run. Hence concepts of joint, marginal, and conditional probability distributions stem from related concepts of frequency distributions. You are interested in investigating the relationship between the age of heads of households and weekly earnings of households. The accompanying data gives the number of occurrences grouped by age and income. You collect data from 1,744 individuals and think of these individuals as a population that you want to describe, rather than a sample from which you want to infer behavior of a larger population. After sorting the data, you generate the accompanying table: J $\text { Joint Absolute Frequencies of Age and Income, 1,744 Households }$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Age of head of household }$ 1 2 3 4 5 Household Income 16 -under 20 20 -under 25 25 -under 45 45 -under 65 65 and > \ 0 -under \ 200 80 76 130 86 24 \ 200 -under \ 400 13 90 346 140 8 \ 400 -under \ 600 0 19 251 101 6 \ 600 -under \ 800 1 11 110 55 1 \ 800 and > 1 1 108 84 2 The median of the income group of $800 and above is $1,050. (a)Calculate the joint relative frequencies and the marginal relative frequencies. Interpret one of each of these. Sketch the cumulative income distribution. (b)Calculate the conditional relative income frequencies for the two age categories 16-under 20, and 45-under 65. Calculate the mean household income for both age categories. (c)If household income and age of head of household were independently distributed, what would you expect these two conditional relative income distributions to look like? Are they similar here? (d)Your textbook has given you a primary definition of independence that does not involve conditional relative frequency distributions. What is that definition? Do you think that age and income are independent here, using this definition?

Think of the situation of rolling two dice and let M denote the sum of the number of dots on the two dice. (So M is a number between 1 and 12.) (a)In a table, list all of the possible outcomes for the random variable M together with its probability distribution and cumulative probability distribution. Sketch both distributions. (b)Calculate the expected value and the standard deviation for M. (c)Looking at the sketch of the probability distribution, you notice that it resembles a normal distribution. Should you be able to use the standard normal distribution to calculate probabilities of events? Why or why not?

If variables with a multivariate normal distribution have covariances that equal zero, then

Assume that Y is normally distributed N(μ, σ²). Moving from the mean (μ)1.96 standard deviations to the left and 1.96 standard deviations to the right, then the area under the normal p.d.f. is

Show in a scatterplot what the relationship between two variables X and Y would look like if there was (a)a strong negative correlation. (b)a strong positive correlation. (c)no correlation.

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.

SAT scores in Mathematics are normally distributed with a mean of 500 and a standard deviation of 100. The formula for the normal distribution is $f(Y)=\frac{1}{\sqrt{2 \pi \sigma_{Y}^{2}}} e^{-\frac{1}{2}\left(\frac{Y-\mu Y}{\sigma Y}\right)^{2}}$ Use the scatter plot option in a standard spreadsheet program, such as Excel, to plot the Mathematics SAT distribution using this formula. Start by entering 300 as the first SAT score in the first column (the lowest score you can get in the mathematics section as long as you fill in your name correctly), and then increment the scores by 10 until you reach 800. In the second column, use the formula for the normal distribution and calculate f(Y). Then use the scatter plot option, where you eventually remove markers and substitute these with the solid line option.

You consider visiting Montreal during the break between terms in January. You go to the relevant Web site of the official tourist office to figure out the type of clothes you should take on the trip. The site lists that the average high during January is -7° C, with a standard deviation of 4° C. Unfortunately you are more familiar with Fahrenheit than with Celsius, but find that the two are related by the following linear function: C= $\frac { 5 } { 9 }$ (F - 32). Find the mean and standard deviation for the January temperature in Montreal in Fahrenheit.

Using the fact that the standardized variable Z is a linear transformation of the normally distributed random variable Y, derive the expected value and variance of Z.

Use a standard spreadsheet program, such as Excel, to find the following probabilities from various distributions analyzed in the current chapter: a. If Y is distributed , find Pr(Y ? 7.78) b. If Y is distributed $X _ { \mathrm {10 } } ^ { 2 }$ , find Pr(Y > 18.31) c. If Y is distributed F_10,?, find Pr(Y > 1.83) d. If Y is distributed t₁₅, find Pr(Y > 1.75) e. If Y is distributed t₉₀, find Pr(-1.99 ?Y ? 1.99) f. If Y is distributed N(0,1), find Pr(-1.99 ?Y ? 1.99) g. If Y is distributed F_10,4, find Pr(Y > 4.12) h. If Y is distributed F_7,120, find Pr(Y > 2.79)

The accompanying table shows the joint distribution between the change of the unemployment rate in an election year and the share of the candidate of the incumbent party since 1928. You think of this data as a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population. Joint Distribution of Unemployment Rate Change and Incumbent Party's Vote Share in Total Vote Cast for the Two Major-Party Candidates, 1928-2000 ( Incumbent -50\%)>0 (Y=0) ( Incumbent -50\%)\leq0 (Y=1) Total \Deltau>0(X=0) 0.053 0.211 0.264 \Deltau\leq0(X=1) 0.579 0.157 0.736 Total 0.632 0.368 1.00 (a)Compute and interpret E(Y)and E(X). (b)Calculate E(Y $\mid X$ = 1)and E(Y $\mid X$ = 0). Did you expect these to be very different? (c)What is the probability that the unemployment rate decreases in an election year? (d)Conditional on the unemployment rate decreasing, what is the probability that an incumbent will lose the election? (e)What would the joint distribution look like under independence?

You are at a college of roughly 1,000 students and obtain data from the entire freshman class (250 students)on height and weight during orientation. You consider this to be a population that you want to describe, rather than a sample from which you want to infer general relationships in a larger population. Weight (Y)is measured in pounds and height (X)is measured in inches. You calculate the following sums: $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 }$ = 94,228.8, $\sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 }$ = 1,248.9, $\sum _ { i = 1 } ^ { n } x _ { i } y _ { i }$ = 7,625.9 (small letters refer to deviations from means as in $\mathrm { z } _ { i }$ = $Z _ { i }$ - $\bar { Z }$ ). (a)Given your general knowledge about human height and weight of a given age, what can you say about the shape of the two distributions? (b)What is the correlation coefficient between height and weight here?

Following Alfred Nobel's will, there are five Nobel Prizes awarded each year. These are for outstanding achievements in Chemistry, Physics, Physiology or Medicine, Literature, and Peace. In 1968, the Bank of Sweden added a prize in Economic Sciences in memory of Alfred Nobel. You think of the data as describing a population, rather than a sample from which you want to infer behavior of a larger population. The accompanying table lists the joint probability distribution between recipients in economics and the other five prizes, and the citizenship of the recipients, based on the 1969-2001 period. Joint Distribution of Nobel Prize Winners in Economics and Non-Economics Disciplines, and Citizenship, 1969-2001 U.S. Citizen (Y=0) Non= U.S. Citizen (Y=1) Total Economics Nobel Prize (X=0) 0.118 0.049 0.167 Physics, Chemistry, Medicine, Literature, and Peace Nobel Prize (X=1) 0.345 0.488 0.833 Total 0.463 0.537 1.00 (a)Compute E(Y)and interpret the resulting number. (b)Calculate and interpret E(Y $\mid X$ =1)and E(Y $\mid X$ =0). (c)A randomly selected Nobel Prize winner reports that he is a non-U.S. citizen. What is the probability that this genius has won the Economics Nobel Prize? A Nobel Prize in the other five disciplines? (d)Show what the joint distribution would look like if the two categories were independent.