Due (10^th Dec, 11:45p.m).    Instructions

Question 1- Random variables

1. a) Suppose you toss a fair coin (equally likely to turn up heads or tails) four times. This procedure describes random events. What is the sample space for this procedure?

(2 points)

1. b) If we define a random variable that indicates the sum of the number of heads (in the procedure from the previous part), what is the expected value of this random variable?

(2 points)

1. c) If the coin (in the procedure from the previous part) is not fair and turns up heads a proportion p of the time (p is not 0.5), what is the expected number of heads?

(2 points)

1. d) When I lived in Toronto, in the Fall I biked home from work in the dark. My route had 4 potholes on it and I hit all four each time. I’d start my commute with my bike light turned on, but it had a loose electrical connection so each time I hit a pothole with probability p the light would switch from on to off or from off to on. What was the probability that my light would be turned on after my commute?

(2 points)

1. e) Similar to the previous part, assume that the number of potholes varied and the probability that I hit k potholes during my commute was λ^k expp´λq{k!, for a fixed λ ą This is the probability distribution function of a Poisson. This distribution arises when potholes occur independently with rate λ. Suppose that the probability that my light is switched (to off if on, to on if off) over each pothole was now p “ 1.

 
 
What was the probability that my light would be turned on after the commute, in terms of λ? (As before, my commute began with the light on). You may express your answer using trigonometric or special functions: Do not leave it as an infinite series.
(2 points)

Question 2- Multiple choice  (Show work for multiple choice questions)

1. In a study by Dr. Ryan W. Allen et al. (Environmental Pollution, 2019), air quality was assessed in 342 apartments in Ulaanbaatar, Mongolia. A total of 87 explanatory variables (‘predictor variables obtained from outdoor monitoring data, questionnaires, home assessments, and geographic data sets’; Environmental

 
Pollution, 2019) were used to predict the air quality. If p-values were computed for each of the explanatory variables, what α level would be best for assessing significance?

1. α “01
2. α “05
3. α “46E-4
4. α “75E-4
5. α “68E-6
6. Not enough information is given to provide an informed decision.
   • points)
7. On November 9th 2020, Pfizer announced interim results for a Phase 3 study on the COVID-19 vaccine BNT162b2. Their study enrolled 43,538 subjects divided into a control arm and a study arm. Their interim results indicated 90% vaccine efficacy after 94 infection events among the study subjects. What statistical tool would be most appropriate for assessing the significance of this result?
   1. An unpaired two-sample t-test.
   2. Pearson’s correlation.
   3. A log-rank test.
   4. An F-test.
   5. Fisher’s exact test.
   6. Kaplan-Meier curves.
      • points)