Download SIO 221B Homework #1 1. During the professional basketball

SIO 221B
Homework #1
1. During the professional basketball seasons that ended in 2016, the best free-throw shooter in
the NBA was Stephen Curry who made 90.8% of his free throws. The best free-throw shooter in
the WNBA was Shenise Johnson who made 93.8% of her free throws. Men and women use the
same basketball hoop, which has a diameter of 18 inches. The men’s basketball is 9.55 inches in
diameter, while the women’s basketball is smaller at 9.23 inches in diameter.
a. Who was the more accurate free-throw shooter, Curry or Johnson? In answering this
question, assume that their shooting can be modeled as a two-dimensional isotropic
normal joint probability density function whose mean is the center of the hoop. Accuracy
can be expressed as the standard deviation of the normal joint probability density
function.
b. If their accuracy stays the same while shooting a different size ball, what percentage of
free throws would each make after switching balls?
2. Consider two independent random variables x and y, uniformly distributed between 0 and a.
What is the probability density function of the variable z = xy? Hint: First, find the cumulative
distribution function of z.
3. Consider the function
⎧ c, 2 < r < 4
⎪
! h ( r ) = ⎨ c, −1 < r < 1 ,
⎪ 0, elsewhere
⎩
where r is a real variable, and c is a constant.
a. What must the value of c be for h(r) to be a valid probability density function.
b. Calculate the first two moments of h(r).
c. Using a computer, generate random variables governed by h(r) and plot a histogram.
Hint: use the rand function in Matlab.
d. By summing a number of these random variables, show the approach to a normal
distribution as suggested by the central limit theorem. Plot the pdfs of sums of 2, 10, and
100 random variables.
4. Generate joint-normally-distributed variables x and y such that ! x = y = 0 , ! x 2 = 2 ,
! y 2 = 0.25 , and ! xy = 0.5 . Do this by generating two independent normally distributed
variables, then scaling and rotating them.
a. Make a scatter plot of 100,000 realizations of x and y.
b. Calculate and plot the joint pdf from these same 100,000 realizations.
c. By direct calculation from these 100,000 realizations prove that the means, variances, and
covariance of x and y are correct.