ECON83A - Statistics for Economic Analysis Problem Set 4

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ECON83A - Statistics for Economic Analysis

Problem Set 4

C4 “Discrete Random Variables and Probability Distributions”

Please show all the work necessary to explain why your answer is correct!

1.  Suppose that the number of M&Ms in each snack size bag has the following probability density function:

Number of M&Ms     47       48      49      50      51     52      53

Probability               0.05    0.15    0.2    0.25    0.2    0.1    0.05

(a) What is the cumulative probability density function? Please graph it. (Specifically, please put number the

of M&Ms on the horizontal axis and the cumulative probability on the vertical axis.) (b) What is the average number of M&Ms in a snack size bag?

(c) What is the probability that a randomly selected snack size M&Ms bag will contain at least 49 but no more than 51 pieces?

(d) If two packages are independently selected at random, what is the probability that at least one of them contains at least 50 pieces?

2.  Suppose that you apply to 6 top MBA programs and that your probability of getting into each program is 20%. You may assume that your chances of getting into one program are independent of your chances of getting into another program.

(a) What is probability that you’ll get into all the programs?

(b) What is the probability that you’ll get into no programs?

(c) What is the probability that you’ll get into at least 2 programs?

(d) What does this question have to do with the binomial distribution? Please be specific about p and n.

3.  Suppose that if you go fishing on Massell Pond, you can expect to catch about 4 fish a day on average. If the distribution of fish catching is Poisson,

(a) What is probability that you’ll catch fewer than 3 fish?

(b) What is the probability that you’ll catch 3 or more fish?

(c) What is the probability that you’ll catch 4 or more fish?

(d) What is the probability that you’ll catch 5 or more fish?

4.  Suppose that the lacrosse team has 15 league games per season and a 70% chance of winning each game.

(a) Using R, calculate the probability distribution function (PDF) for x = 0, 1, . . . , 15.

i.  Graph the PDF. Please provide a title and label the axes.

ii. Is the PDF symmetric or skewed? Why?

(b) Plot the cumulative probability distribution function (CDF) for x = 0, 1, . . . , 15.

5. Consider the following table showing the joint probability of the value of the S&P 500 and per capita GDP:

 

Y = Value of the S&P 500 $8,000          $10,000

 

 

X = Per Capita GDP

$0        $25,000 $50,000

 

0.2

0.1

0.1

0.1

0.2

0.3

0.3

0.3

0.4

 

0.4                 0.6

 

(a) What is the expected value of the S&P 500?

(b) What is the standard deviation of the S&P 500?

(c) Is per capita GDP independent of the value of the S&P 500?

(d) What is the covariance of per capita GDP and the value of the S&P 500?









2022-10-12