Mat 301
Review Exercises for Exam 2
1. Use R to compute the probability of drawing a random variable between 0.05 and 1.2.
The PDF is:
2. Given an arbitrary PDF, p(x) defined its CDF.
3. Assign the CDF expressions for F(b), F(b)-F(a), F(a) to the proper part of the figure:
4. Let Z be a standard normal random variable. Sketch a graph for each and calculate
Pr(0 ≤ Z ≤ 2.94)
Pr (0 ≤ Z ≤ 1)
Pr (−3.723 ≤ Z ≤ 0)
Pr (−4.70 ≤ Z ≤ 4.70)
Pr (Z ≤ 1.32)
Pr (−0.75 ≤ Z)
Pr (−1.76 ≤ Z ≤ 2.00)
Pr (1.3746 ≤ Z ≤ 2.5035)
Pr (1.71 ≤ Z)
Pr (|Z| ≤ 2.50)
5. For Pr(c ≤ Z) = 0.278, what is general name c is called and what value should it be?
6. 4.6 is a N(3.5, 1.2) random variable. What is its standard value?
7. Compute the expected value and variance of a Weibull distribution that has shape
parameter 3.1 and scale parameter of 0.89.
8. Rowe and Hanson (FSI 28(3-4):239-250) found the following data for the square root
of the smallest rectangular area containing a shotgun pellet pattern at 10m:
6.84, 6.32, 6.96, 5.85, 5.95, 6.29
Calculate a point estimate for the mean, median, 48th percentile and Pr(sqrt-area < 6.32).
9. The data in question 8 is believed to follow a gamma distribution. Find estimates of the
shape and scale parameters with the method of moments estimators for the gamma
distribution.
10. Some data is believed to follow a log-normal distribution. The log–normal probability
density, first and second moments are:
a. Show the integrals required to evaluate the first and second moments. Do not actually
attempt the integration.
b. Using the moment equations determine a set of estimators for  and  (two equations).
c. The data below is believed to follow a log normal distribution. Use the estimators
found above to compute the mean and standard deviation estimates.
18.5351272, 43.7957107, 0.4053501, 0.5144618, 7.6332742, 18.0592203, 15.5749319,
26.3539858, 13.8881712, 2.5133146
11. Consider the data thought to follow a Gaussian distribution:
33.1068240, -6.3331469, 22.2903011,
29.2973275, -34.2364631, -4.4113930,
-6.2030667, 0.3504070 , -19.4854146,
17.0975034, -6.8416016, 0.8700698,
-23.7861226, -1.8070511, 8.9497832,
30.4888656, 8.3810280, -12.4548588,
-6.9142991, -21.7899524,
-25.7845033, -12.0375226,
-10.5979710, -27.1274107,
-9.9103781, 8.3101956,
27.5315404, 5.9680535,
-19.5759143
-12.1741625,
17.9397860,
20.4247578,
-6.9263659,
-2.4057959,
a. Compute the maximum likelihood estimates for the mean and standard deviation.
b. Using these estimates for m and s estimate the sampling distributions for these
parameters using the parametric bootstrap with 2000 sampling iterations. What are the
means of the bootstrap mean and standard deviation estimates? How do they compare to
the maximum likelihood estimates?
c. Compute the two-sided 80%, 95% and 99% confidence intervals for the mean. Use the
appropriate sample size procedure.