Exam 2: Review of Probability

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 ,find Pr(Y > 18.31) c.If Y is distributed F10,∞,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 F10,4,find Pr(Y > 4.12) h.If Y is distributed F7,120,find Pr(Y > 2.79)

Explain why the two probabilities are identical for the standard normal distribution: Pr(-1.96 ≤ X ≤ 1.96)and Pr(-1.96 < X < 1.96).

The systolic blood pressure of females in their 20s is normally distributed with a mean of 120 with a standard deviation of 9.What is the probability of finding a female with a blood pressure of less than 100? More than 135? Between 105 and 123? You visit the women's soccer team on campus,and find that the average blood pressure of the 25 members is 114.Is it likely that this group of women came from the same population?

Find the following probabilities: (a)Y is distributed .Find Pr(Y > 9.49). (b)Y is distributed t∞.Find Pr(Y > -0.5). (c)Y is distributed F4,∞.Find Pr(Y < 3.32). (d)Y is distributed N(500,10000).Find Pr(Y > 696 or Y < 304).

The correlation between X and Y

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= (F - 32). Find the mean and standard deviation for the January temperature in Montreal in Fahrenheit.

The Economic Report of the President gives the following age distribution of the United States population for the year 2000: United States Population By Age Group,2000 Imagine that every person was assigned a unique number between 1 and 275,372,000 (the total population in 2000).If you generated a random number,what would be the probability that you had drawn someone older than 65 or under 16? Treating the percentages as probabilities,write down the cumulative probability distribution.What is the probability of drawing someone who is 24 years or younger?

Consider the following linear transformation of a random variable y = where μx is the mean of x and σx is the standard deviation.Then the expected value and the standard deviation of Y are given as

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

In considering the purchase of a certain stock,you attach the following probabilities to possible changes in the stock price over the next year. What is the expected value,the variance,and the standard deviation? Which is the most likely outcome? Sketch the cumulative distribution function.

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: = 94,228.8, = 1,248.9, = 7,625.9 (small letters refer to deviations from means as in = - ). (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?

The conditional distribution of Y given X = x,Pr(Y = y =x),is

From the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website the chapter 8 CPS data set (ch8_cps.xls)into a spreadsheet program such as Excel.For the exercise,use the first 500 observations only.Using data for average hourly earnings only (ahe),describe the earnings distribution.Use summary statistics,such as the mean,median,variance,and skewness.Produce a frequency distribution ("histogram")using reasonable earnings class sizes.

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?

There are frequently situations where you have information on the conditional distribution of Y given X,but are interested in the conditional distribution of X given Y.Recalling Pr(Y = y = x)= ,derive a relationship between Pr(X = x = y)and Pr(Y = y = x).This is called Bayes' theorem.

The cumulative probability distribution shows the probability

The textbook formula for the variance of the discrete random variable Y is given as Another commonly used formulation is . Prove that the two formulas are the same.

The mean and variance of a Bernoille random variable are given as

The expectations augmented Phillips curve postulates △p = π - f (u - ), where △p is the actual inflation rate,π is the expected inflation rate,and u is the unemployment rate,with "-" indicating equilibrium (the NAIRU - Non-Accelerating Inflation Rate of Unemployment).Under the assumption of static expectations (π = △p-1),i.e. ,that you expect this period's inflation rate to hold for the next period ("the sun shines today,it will shine tomorrow"),then the prediction is that inflation will accelerate if the unemployment rate is below its equilibrium level.The accompanying table below displays information on accelerating annual inflation and unemployment rate differences from the equilibrium rate (cyclical unemployment),where the latter is approximated by a five-year moving average.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.The data is collected from United States quarterly data for the period 1964:1 to 1995:4. Joint Distribution of Accelerating Inflation and Cyclical Unemployment, 1964:1-1995:4 (a)Compute E(Y)and E(X),and interpret both numbers. (b)Calculate E(Y = 1)and E(Y = 0).If there was independence between cyclical unemployment and acceleration in the inflation rate,what would you expect the relationship between the two expected values to be? Given that the two means are different,is this sufficient to assume that the two variables are independent? (c)What is the probability of inflation to increase if there is positive cyclical unemployment? Negative cyclical unemployment? (d)You randomly select one of the 59 quarters when there was positive cyclical unemployment ((u - )> 0).What is the probability there was decelerating inflation during that quarter?