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ECMT3160 Statistical Modelling

TAKE-HOME FINAL EXAMINATION

SEMESTER 1, 2021

Questions.

Question 1. (4 Marks in total; 2 marks per part)

Compute the mean and the variance of the following probability distribution:

1. p(x) = λx入 − 1 , 0 < x < 1, λ > 0.

2. p(x) = (x - 1)2 , 0 < x < 2.

Question 2. (10 Marks in total) .

Let Υ , Ψ , and Ξ be random variables such that

Ξ = Ψ/8 and

Υ = ,^-

if Ψ > 0, otherwise

where Ψ is uniformly distributed on the interval ( -1, 1). What is the probability that the equation Υx2 + Ψx + Ξ = 0

has solutions in the real number system?

Question 3. (6 Marks in total) .

After the ﬁrst quarter of a president’s term, a poll indicated that a near record (low) level of 40% of eligible voters expressed “a great deal” of conﬁdence in his government. You decide to conduct your own telephone survey at that time and randomly called people in your district and asked them to describe their level of conﬁdence in the government.

1. You decide to keep calling until the ﬁrst call at which there are k people who have expressed “a great deal” of conﬁdence in the government. Find the probability of making n calls. [2 marks]

2. You believe that p, the proportion of people who have a great deal of conﬁdence in the government, is very much larger than 40% in your district. That is, p → 1 is expected in your district. Consequently, you decide to increase k in such a way that (1 - p)k = 0.4, as p → 1. Find the probability of making n + k calls as p → 1, and interpret this result. [4 marks]