Interactive and Computational Probability
|Department of Statistics
Instructor: Ivo Dinov
|Due Date: Monday, Mar. 14, 2005
- (HW_5_1) Consider
writing onto a computer disk and then sending it through a certifier
counts X, the number of missing pulses. Suppose this number X has a
distribution with parameter λ = 0.2. Run the Poisson Experiment under the SOCR Experiments
and calcualte emperically (by running 100 or more experiments) and
theoretically (by using the Poisson probability formula) each of the
(a) The probability that a disk has exactly one missing pulse?
(b) The probability that a disk has at least two missing pulses?
(c) If two disks are independently selected, what is the probability
that neither contains a missing pulse?
- (HW_5_2) The Pareto family of distributions (see SOCR Distributions) has been used to approximate the distribution of income,
population size, size of firms, among others. The Pareto distribution has
parameters, k and ϑ, both strictly positive, and its pdf is
(a) Describe the effects of the two parameters on the shape of the graph of f(x; k, ϑ).
(b) Verify that the total area under the for all equals 1.
(c) If the R.V. X has Pareto Distribution f(x; k, ϑ) and
b > ϑ obtain an expression for the CDF F(b) = P(X ≤
(d) For ϑ < a < b, obtain an expression for the probability
P(a ≤ X ≤ b) = F(b) − F(a).
(e) If k > 1, compute E(X).
(f) What are the expected value, E(X), and the standard deviation, SD(X), if k = 1 and ϑ=4? Validate your closed-form mathematical expression by using the Pareto Distribution Modeler.
- (HW_5_3) Suppose the diameter at
breast height of trees (in.) of a certain type is Normally distributed with µ = 8.8
= 2.8. Use the Normal Distribution Modeler part of SOCR Distributions to compute the following probabilities.
(a) What is the probability that the diameter of a randomly selected
tree will be at least 10 in.? Will exceed 10 in.?
(b) What is the probability that the diameter of a randomly selected
tree will exceed 20 in.?
(c) What is the probability that the diameter of a randomly selected
tree will be between 5 and 10 in.?
(d) What value c is such that the interval (8.8−c,
8.8+c) includes 98% of all diameter values?
- (HW_5_4) The Rockwell hardness of a
is determined by impressing a hardened point into the surface of the
and then measuring the depth of penetration of the point. Suppose the
hardness of a particular alloy is measured on a continuous
scale and is Normally distributed with mean 70 and
deviation 3. Use the Normal Distribution Modeler part of SOCR Distributions to compute the following:
(a) If a specimen is acceptable only if its hardness is between 67 and
what is the chance that a randomly chosen specimen has an
(b) If the acceptable range of hardness is (70−c, 70+c),
for what value of c would 95% of all specimens have
(c) If the acceptable range is as in part (a) and the hardness of each
of ten randomly selected specimens is independently determined, what is
expected number of acceptable specimens among the ten?