Department of Mathematical and Statistical Sciences
Stat 366 Asg 1
Due on Sept 22 2006
1. Of the travelers arriving at a small airport, 60% fly on major airlines, 30% fly on privately owned
planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those
traveling on major airlines, 50% are traveling for business reasons, whereas 60% of those arriving on
private planes and 90% of those arriving on other commercially owned planes are travelling for business
reasons. Suppose that we randomly select one person arriving at this airport. What is the probability
that the person
(a) is travelling on business?
(b) is travelling for business on a privately owned plane?
(c) arrived on a privately owned plane, given that the person is travelling for business reasons?
2. A salesperson can contact either one or two customers per day with probability 1/3 and 2/3,respec-
tively. Each contact will result in either no sale or a $100 sale, with the probabilities 0.8 and 0.2,
respectively. Find the probability distribution for daily sales. Find the mean and variance of the daily
sales.
3. The length of time Yto complete a key operation in the construction of houses has an exponential
distribution given by f(y) = 1
10 e−y/10,y > 0. The formula C= 40Y+ 100 relates the cost of
completing this operation to the time to completion Y. Find the mean and variance of C. Would you
expect Cto exceed 2000 hours very often? Explain.
4. Suppose that Y1, Y2, ...., Y40 denote a random sample of measurements on the proportion of impurities
in iron ore samples. Let each variable Yihave a probability density function given by
f(y) = 12y2(1 −y),0≤y≤1.
The ore is to be rejected by the potential buyer if the sample mean Yexceeds 0.7.
(i) Find the probability of rejecting the ore by the buyer.
(ii) If 10 different samples of size 40 are taken from iron ore production, what is the probability that
at least 3 of them will have to be rejected?
5. Let Y1, Y2, ...., Yndenote a random sample from a Poisson distribution with mean λ.
(i) Suggest an unbiased estimator for λ.
(ii) Define C= 3Y+Y2.Show that E(C) = 4λ+λ2.
(iii) Find a function of Y1, Y2, ...., Ynthat is an unbiased estimator of E(C).
6. Let Y1, Y2, ...., Yndenote a random sample of size nfrom a population with whose density is given by
f(y) = 5y4
θ5,0≤y≤θ,
where θ > 0 is unknown. Consider two estimators b
θ1=6
5Yand b
θ2=5n+1
5nY(n),where Y=1
nPYi
and Y(n)= max (Y1, Y2, ...., Yn).Find the efficiency of b
θ1relative to b
θ2.Which one do you recommend?
7. Let Y1, Y2, ...., Yndenote a random sample of size nfrom a population with whose density is given by
f(y) = 3θ3
y4, y ≥θ,
where θ > 0 is unknown. Consider the estimator b
θ= min (Y1, Y2, ...., Yn).Find the mean squared error
(MSE) of b
θ.
8. Let Y1, Y2, ...., Yndenote a random sample of size nfrom a normal distribution with mean θ1and
variance θ2.Find the maximum likelihood estimators of θ1,θ2and θ1
θ2.
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