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STA237 (Probability, Statistics and Data Analysis I) - Fall 2021

Assignment 1 

 

 

1.  Given independent random variables X and y , with means and standard deviations as shown,

 

 

Mean

SD

X

y

120

300

12

16

find the mean and standard deviation of each of the variables below. Also, y1 . y2 . y3 . and y4  are independent variables with the same distribution as y .

(a) 0z8y

(b)  2X − 100

(c) X + 2y

(d) 3X − y

(e) y1 + y2 + y3 + y4

2. A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment.  Previous research had established that this trait is usually found in one of every eight frogs. He collects and examines a dozen frogs. If the frequency of the trait has not changed, what is the probability that he finds the trait in

(a) none of the 12 frogs?

(b) at least two frogs?

(c) three or four frogs?

(d) no more than four frogs?

3.  Scores on an examination are assumed to be normally distributed with mean 78 and variance 36.

(a) What is the probability that a person taking the examination scores higher than 72?

(b)  Suppose that students scoring in the top 10% of this distribution are to receive an A grade.  What is the minimum score a student must achieve to earn an A grade?

(c) What must be the cutoff point for passing the examination if the examiner wants only the top 28.1% of all scores to be passing?

(d) If it is known that a student’s score exceeds 72, what is the probability that his or her score exceeds 84?

4. The number of ways you can choose r things from a set of n, ignoring the order in which they are chosen, is  ╱  r(n)  = . Let z be the first element of the set of n things. We can partition the collection of possible size r subsets into those that contain z and those that don’t: there must be ╱subsets of the first type and ╱nr(−)1subsets of the second type. Thus

╱  r(n)    =   + ╱nr(−) 1z

Using this and the fact that  ╱  n(n)  = ╱  0(n)  = 1, write a recursive function to calculate ╱  r(n) .

5. The following dice game cost ✩10 to play.  If you roll 1, 2, or 3, you lose your money. If you roll 4 or 5, you get your money back. If you roll a 6, you win ✩24.

(a) Find the probability mass function of your winnings W .

(b) Find the expected value of the game.

(c)  Simulate this dice game. Estimate the expectation and variance of your winnings.