1. A mail-order computer business has six telephone lines. Let X ...

c. What is the probability that observed depth is at most 10? Between
10 and 15?
d. What is the probability that the observed depth is within 1 SD
of the mean value? Within 2 SD’s?
10. Let X have the Pareto pdf given by
kθk f(x; k, θ) = xk+1 x ≥ θ
0 x < θ
a. If k > 1 , compute E(X).
b. What can you say about E(X) if k = 1 ?
c. If k > 2, show that V (X) = kθ 2 (k − 1) −2 (k − 2) −1
d. If k = 2, what can you say about V (X)?
e. What conditions on k are necessary to ensure that E(X n ) is finite?
11. If bearing diameter is normally distributed, what is the probability
that the diameter of a randomly selected bearing is
a. Within 1.5 SD’s of its mean value? (Φ(1.5)−Φ(−1.5) = 0.8663856)
b. Farther than 2.5 SD’d from its mean value? (1 − [Φ(2.5) −
Φ(−2.5)]= 1 − 0.9875807= 0.0124193)
c. Between 1 and 2 SD’s from its mean value? ([Φ(2) − Φ(1)] +
[Φ(−1) − Φ(−2)] = 0.2718102)
12. The inside diameter of a randomly selected piston ring is a r.v. with
mean value 12 cm and standard deviation 0.04 cm.
a. If ¯ X is the sample of n = 16 rings, where is the sampling distribution
of ¯ X centred, and what is the standard deviation of the
¯X distribution?
Ans ¯ X is centred at 12, V [ ¯ X] = σ 2 /n = (0.04) 2 /16 or, s.d. is
0.04/4 = 0.01.
b. Answer the questions posed in part (a) for a sample of size n = 64
rings.
Ans ¯ X is centred at 12, V [ ¯ X] = σ 2 /n = (0.04) 2 /64 or, s.d. is
0.04/8 = 0.005.
c. For which of the two random samples, the one of part (a), or the
one of part (b), is ¯ X more likely to be within 0.01 cm of 12 cm?
Explain your reasoning.
Ans part (b), because, (b)has smaller variance/s.d.
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