Exam 2: Review of Probability

The variance of $\bar { Y } , \sigma _ { \bar { Y } } ^ { 2 }$ is given by the following formula:

Calculate the following probabilities using the standard normal distribution.Sketch the probability distribution in each case, shading in the area of the calculated probability. (a) $\operatorname { Pr } ( Z < 0.0 )$ (b) $\operatorname { Pr } ( Z \leq 1.0 )$ (c) $\operatorname { Pr } ( Z > 1.96 )$ (d) $\operatorname { Pr } ( Z < - 2.0 )$ (e) $\operatorname { Pr } ( Z > 1.645 )$ (f) $\operatorname { Pr } ( Z > - 1.645 )$ (g) $\operatorname { Pr } ( - 1.96 < Z < 1.96 )$ (h) $\operatorname { Pr } ( Z < 2.576$ or $Z > 2.576 )$ (i) $\operatorname { Pr } ( Z > z ) = 0.10$ ; find $z$ . (j) $\operatorname { Pr } ( Z < - z$ or $Z > z ) = 0.05$ ; find $z$ .

In considering the purchase of a certain stock, you attach the following probabilities to possible changes in the stock price over the next year. Stock Price Change During Next Twelve Months (\%) Probability +15 0.2 +5 0.3 0 0.4 -5 0.05 -15 0.05 What is the expected value, the variance, and the standard deviation? Which is the most likely outcome? Sketch the cumulative distribution function. .

The conditional expectation of Y given $X , E ( Y \mid X = x )$ is calculated as follows:

The expectations augmented Phillips curve postulates $\Delta p = \pi - f ( u - \bar { u } ) ,$ where $\Delta p$ is the actual inflation rate, $\pi$ 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 $\left( \pi = \Delta p _ { - 1 } \right)$ , 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 (u-)>0 (u-)\geq0 Total (Y=0) (Y=1) \Deltap-\Delta>0 0.156 0.383 0.539 (X=0) \Deltap-\Delta\leq0 0.297 0.164 0.461 (X=1) Total 0.453 0.547 1.00 (a)Compute EY( )and EX( ), and interpret both numbers.

Assume that Y is normally distributed $N \left( \mu , \sigma ^ { 2 } \right)$ Moving from the mean $( \mu )$ 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

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: 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.

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 Outcome (age category) Under 5 5-15 16-19 20-24 25-44 45-64 65 and over Percentage 0.06 0.16 0.06 0.07 0.30 0.22 0.13 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?

The conditional distribution of Y given $X = x , \operatorname { Pr } ( Y = y \mid X = x )$ is

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 $\operatorname { Pr } ( Y = y \mid X = x ) = \frac { \operatorname { Pr } ( X = x , Y = y ) } { \operatorname { Pr } ( X = x ) }$ , derive a relationship between $\operatorname { Pr } ( X = x \mid Y = y ) \text { and } \operatorname { Pr } ( Y = y \mid X = x )$ This is called Bayes' theorem.

Let Y be a random variable.Then var(Y)equals a. $\sqrt { E \left[ \left( Y - \mu _ { Y } \right) ^ { 2 } \right] }$ . b. $E \left[ \left| \left( Y - \mu _ { Y } \right) \right| \right]$ . c. $E \left[ \left( Y - \mu _ { Y } \right) ^ { 2 } \right]$ . d. $E \left[ \left( Y - \mu _ { Y } \right) \right]$ .

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

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 -70 C, with a standard deviation of 40 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.

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: $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 94,228.8 , \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } = 1,248.9 , \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } = 7,625.9$ $\text { (small letters refer to deviations from means as in } z _ { i } = Z _ { i } - \bar { Z } \text { ). }$ (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?

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 $X_{1}$ and $X_{2}$ 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 $X_{1}$ and $X_{2}$ . For a catch-up effect to be present, what relationship must the two standard deviations show? Is this the case 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 EY( )and interpret the resulting number.

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?

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.

Two variables are uncorrelated in all of the cases below, with the exception of a. being independent. b. having a zero covariance. c. $\left| \sigma _ { X Y } \right| \leq \sqrt { \sigma _ { X } ^ { 2 } \sigma _ { Y } ^ { 2 } }$ . d. $E ( Y \mid X ) = 0$ .

In econometrics, we typically do not rely on exact or finite sample distributions because a. we have approximately an infinite number of observations (think of re-sampling). b. variables typically are normally distributed. c. the covariances of $Y _ { i } , Y _ { j }$ are typically not zero. d. asymptotic distributions can be counted on to provide good approximations to the exact sampling distribution (given the number of observations available in most cases).