|STAT 110 A
Probability & Statistics for
|Department of Statistics
Instructor: Ivo Dinov
|Due Date: Wednesday, Nov. 24, 2004
Please, submit your homework right before lecture on
the due date. Correct solutions to any 5 out of the 6 problems carry
credit. See the HW
. On the front page include the following
- (HW_4_1) [Sec. 3.5, #69] Suppose that p
P(male birth) = 0.5. A couple wishes to have exactly two female
in their family. The will have children until this condition is
(a) What is the probability that the family has x male children?
(b) What is the probability that the family has four children?
(c) What is the probability that the family has at most four children?
(d) How many male children would you expect this family to have? How
many children would you expect this family to have?
- (HW_4_2) [Sec. 3.6, #76] Consider
writing onto a computer disk and then sending it through a certifier
counts the number of missing pulses. Suppose this number X has a
distribution with parameter λ = 0.2. (Suggested in “Average
Sample Number of Semi-Curtailed Sampling Using the Poisson
Distribution,” J. Quality Technology, 1983, pp. 126-129.
(a) What is the probability that a disk has exactly one missing pulse?
(b) What is 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_4_3) [Sec. 4.1, #2] Suppose the reaction
temperature (in Co) in a certain chemical process has a
uniform distribution with A = −5 and B = 5.
(a) Express the probability density and the cumulative distribution
function in closed forms.
(b) Compute P(X < 0).
(c) Compute P(−2.5 < X < 2.5).
(d) Compute P(−2 ≤ X ≤ 3).
(e) For k satisfying −5 < k < k + 4 < 5, compute P(k < X
< k + 4).
- (HW_4_4) [Sec. 4.1, #10 & 24] A family of
pdf’s that has been used to approximate the distribution of income,
population size, and size of firms is the Pareto family. The family has
parameters, k and ϑ, both strictly positive, and the pdf is
(a) Sketch 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 pdf f(x; k, ϑ) for any fixed
b > ϑ obtain an expression for 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 can you say about E(X) if k = 1?
(g) If k > 2, show that σ2 = V(X) ≡ kϑ2(k
− 1)−2(k − 2)−1.
(h) If k = 2, what can you say about σ2 = V(X)?
- (HW_4_5) [Sec. 4.3, #33] Suppose the diameter at
breast height (in.)
of trees of a certain type is Normally distributed with µ = 8.8
= 2.8, as suggested in the article “Simulating a Harvester-Forwarded
Thinning” (Forest Products J., May 1997: 36-41).
(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_4_6) [Sec. 4.3, #42] 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 Normally distributed with mean 70 and
deviation 3 (and that Rockwell hardness is measured on a continuous
(a) If a specimen is acceptable only if its hardness is between 67 and
what is the probability 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?
(d) What is the probability that at most eight of the ten independently
selected specimens have a hardness of less than 73.84? (Hint: Y = the
among the ten specimens with hardness less than 73.84 is a binomial
What is p?)