# Due Date: Friday, Apr. 11, 2003, turn in after lecture

Correct solutions to any seven problems carry full credit. See the HW submission rules. On the front page include the following header.

• (HW_1_1) The clearness index was determined for the skies over Baghdad for each of the 365 days during a particular year. It was a contribution to the study of the solar radiation of the Baghdad environment. The following table summarized the resulting data.

 Class interval for the clearness index Number of days Relative Frequency Cumulative Relative Frequency 0.16 - 0.20 3 0.008219 0.00822 0.21 - 0.25 5 0.013699 0.02192 0.26 - 0.30 6 0.016438 0.03836 0.31 - 0.35 8 0.021918 0.06027 0.36 - 0.40 12 0.032877 0.09315 0.41 - 0.45 16 0.043836 0.13699 0.46 - 0.50 24 0.065753 0.20274 0.51 - 0.55 39 0.106849 0.30959 0.56 - 0.60 51 0.139726 0.44932 0.61 - 0.65 106 0.290411 0.73973 0.66 - 0.70 84 0.230137 0.96986 0.71 - 0.75 11 0.030137 1.00000

• Determine the Relative Frequency and the Cumulative Relative Frequency (fill in the two last columns of the table)
• Sketch the Relative Frequency histogram and comment on it.
• Cloudy days are those with the clearness index 0.35. What proportion of the days were cloudy?
• Clear days are those for which the clearness index is at least 0.66. What proportion of the days were clear?

• (HW_1_2)  A power network involves three substations A, B and C. Overloads at any of these substations might result in a blackout of the entire system. Past history shown that if substation A alone experiences an overload, then there is 1% chance of a network blackout. For stations B and C alone these percentages are 2% and 3%, respectively. Overloads at two or more substations simultaneously result in a blackout of 5% of the time. During a heat wave there is a 60% chance that substation A alone will experience an overload. For stations B and C these percentages are 20 and 15%, respectively. There is a 5% chance of an overload at two or more substations simultaneously. During a particular heat wave a blackout due to an overload occurred. Find the probability that the overload occurred at
(a) substation A alone
(b) substation B alone
(c) substation C alone
(d) two or more substations simultaneously.

• (HW_1_3) Suppose A, B and C are random events and P is a probability measure.
(a) If P{A} = a and P{B} = b, show that P{A | B} ≥ (a+b-1)/b .
(b) Show that P{A ∪ B | C} = P{A|C} + P{B|C} − P{A ∩ B | C} .
(c) If P{A | B} < P{A} , show that P{B | A} < P{B}.

• (HW_1_4) Suppose the reaction temperature (in Co) in a certain chemical process has a uniform distribution with A = − 5 and B = 5.
(a) Compute P(X < 0).
(b) Compute P( − 2.5 < X < 2.5).
(c) Compute P( − 2 ≤ X ≤ 3).
(d) For k satisfying − 5 < k < k + 4 < 5, compute P(k < X < k + 4).

• (HW_1_5) A computer consulting firm presently has bids out on three projects. Let Ai = { awarded project i } , for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.28, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.05, P(A2 ∩ A3) = 0.07, P(A1 ∩ A2 ∩ A3) = 0.01. Express in words each of the following events and compute the probability of each event.
(a) A1 ∪ A
(b) A'1 ∩ A'2
(c) A1 ∪ A2 ∪ A3
(d) A'1 ∩ A'2 ∩ A'3
(e) A'1 ∩ A'2 ∩ A3
(f) (A'1 ∩ A'2) ∪ A3

• (HW_1_6) Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses. Suppose this number X has a Poisson 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_1_7) A family of pdf ’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and ϑ, both strictly positive, and the pdf is defined by
 (k ϑk) / xk+1 x > ϑ f(x; k, ϑ) = 0 x < ϑ

(a) Sketch the graph of f(x; k, ϑ).
(b) Verify that the total area under the graph (for all values of k and ϑ) equals 1.
(c) If the random variable X has pdf f(x; k, ϑ) for any fixed b > ϑ obtain a closed form expression for F(b) = P(X ≤ b).

• (HW_1_8) Suppose the diameter at breast height (in.) of magnolia trees is normally distributed with µ = 8.8 and σ = 2.8, as suggested in the article “Simulating a Harvester-Forwarded Softwood 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?