EXAM # 3 PLEASE SHOW ALL WORK!

Size: px

Start display at page:

Download "EXAM # 3 PLEASE SHOW ALL WORK!"

Elijah Gibson
5 years ago
Views:

1 Stat 311, Summer 2018 Name EXAM # 3 PLEASE SHOW ALL WORK! Problem Points Grade Total 100

2 1. A socioeconomic study analyzes two discrete random variables in a certain population of households X = number of adult residents and Y = number of child residents. It is found that their joint probability mass function (pmf) p( x, y ) corresponds to a probability table having the structure that appears below, for some parameters and (These ranges are necessary in order for all the probabilities in the table to be between 0 and 1.) X Y (a) Indicate the marginal pmf px ( x ) on the table, and calculate the mean and variance of X. Show all work. (7 pts) (b) Indicate the marginal pmf py ( y ) on the table, and calculate the mean and variance of Y. Show all work. (7 pts) (c) Prove that, regardless of the possible values of and, the covariance between X and Y is equal to zero. Show all work. (7 pts) (d) Despite (c), only under a specific condition on and are X and Y statistically independent. Find it; show all work. (3 pts) (e) Another variable of interest in the study is the total number of residents per household. Calculate the mean and variance of X + Y. Show all work. [Hint: Use (a),(b), and (c).] (3 pts) (f) Specify the smallest and largest values of the variance of X + Y. Explain. (3 pts).

3 2. In this problem, you are asked to prove the expected value property E[ X Y ] E[ X ] E[ Y ] for any two random vvariables X and Y. For simplicity, assume they are continuous with joint pdf (, ) f x y. Then we have the following definitions: marginal pdf of X f ( x) f ( x, y) dy X y marginal pdf of Y f ( y) f ( x, y) dx Y x Expected Value of X Expected Value of Y E[ X ] x f ( x) dx E[ Y] y f ( y) dy x X y Y Expected Value of a real-valued function g( X, Y ) of X and Y E[ g( X, Y)] g( x, y) f ( x, y) dy dx x y (a) Use the preceding definition to express E[ X Y ] as a double integral. (b) Write this double integral as a sum of two double integrals, and reverse the order of integration of the second from dy dx to dx dy over the XY-plane. (c) Apply an elementary property of integrals to rewrite this sum, using the marginal probabilities of X and Y defined above. Briefly explain. (d) Formally justify the conclusion that this is indeed equal to E[ X ] E[ Y ], as claimed.

3. Given the set of all uniformly-distributed, random points within a circle of radius 1. It was proved in lecture that the mean distance of these points to the center of the circle is equal to 2/3.

4 3. Given the set of all uniformly-distributed, random points within a circle of radius 1. It was proved in lecture that the mean distance of these points to the center of the circle is equal to 2/3. In this problem, you are asked to compute the mean distance of these points to any fixed point on the boundary of the circle, using a similar method. First, without loss of generality, we may position the circle so that it is tangent to the X-axis from above. Furthermore, we may rotate the circle, so that the fixed boundary point is located at the origin (0,0), as shown in the figure. The equation of the circle is thus 2 2 x ( y 1) 1. For the set of all random, uniformly-distributed points ( X, Y ) inside this circle, find the mean distance to the boundary point (0,0). That is, find 2 2 E X Y. (a) Determine the joint pdf f ( x, y ) of the uniform distribution over this circle in the XY-plane. Explain. (b) Prove that the equation of the circle in polar coordinates is r 2sin, and specify the range of. Show all work. (c) Set up BUT DO NOT EVALUATE an expression for the expected distance 2 2 E X Y. (d) Evaluate the expression in (b). Show all work. Note: You may use the calculus result 3 cos sin d cos 3 3 C without proof. 2 (X, Y) 1

5 4. The ages X of a population of college undergraduate students are uniformly distributed between 18 and 22 years old, i.e., X ~ Unif (18, 22).. (a) Determine the corresponding pdf f( x ) for any real value of x, and sketch its graph. Show all work. (6 pts) (b) Calculate the probability that the age of a randomly-selected individual student is between 19 and 21 years old. Explain. (6 pts) (c) Calculate the mean and variance 2 of the random variable X. Show all work. (6 pts) (d) Calculate the (approximate) probability that the average age of a sample of n 3 randomly-selected students is between 19 and 21 years old. Show all work. (6 pts) (e) How large would a random sample have to be, in order to reach a 99.9% probability of having its average age lie between 19 and 21 years old? Show all work. (6 pts)

0.24 adults 2. (c) Prove that, regardless of the possible values of and, the covariance between X and Y is equal to zero. Show all work.

0.24 adults 2. (c) Prove that, regardless of the possible values of and, the covariance between X and Y is equal to zero. Show all work. 1 A socioeconomic stud analzes two discrete random variables in a certain population of households = number of adult residents and = number of child residents It is found that their joint probabilit mass