Stat 104 quantitative methods - what is the value of x in

Question

Question 1) Let X be a discrete random variable with PMF (probability mass function) given by

              x²/a,  if x = -3, -2, -1, 0, 1, 2, 3
p_x(x) =
                0,             otherwise

[to understand this notation p(-3) = (-3*-3)/a = 9/a]

a) Find the value of a
b) Calculate E(X)

Question 2) The probability that a cellular phone company kiosk sells X number of new phone contracts per day is shown below.

x	4	5	6	8	10
P(x)	0.2	0.25	0.35	0.10	0.10

a) Find the mean, variance, and standard deviation for this probability distribution.
b) Suppose the kiosk salesperson makes $80/day (8 hours at $10/hour), plus a $25 bonus for each new phone contract sold. What is the mean and variance of the salesperson's daily salary?

Question 3) In a population of students, the number of calculators owned is a random variable X with P(X = 0) = 0.4, P(X = 1) = 0.5, P(X = 2) = 0.1.

a) Find E(x)
b) Find Var(X)

Question 4) You roll two dice.
a) What is the probability of two sixes? Of exactly one 6? Of no sixes?
b) What is the expected number of sixes that will show?

Question 5) We can simulate the expected value result in part (b) above. Follow the following steps in R:
i. Simulate two dice rolls using (use similar code for die2)
die1=sample(1:6,10000,replace=TRUE)
ii. Combine the two dice rolls into a matrix using
dicerolls=cbind(die1,die2)
iii. Each row of dicerolls represents the outcome of rolling two dice. We want to count how many 6's appear each time we roll two dice. We do that as follows.
num6=head(rowSums(dicerolls==6))
iv. Take the mean of the num6 variable and compare it to part (b) above. How does this mean change if we instead use 1000000 rolls?

Question 6) If random variable X has mean μ and variance σ², show (using the a+bX rule) what the mean and variance of Z = (X - μ/)σ are.

Question 7) Find the variance of each of the following bets from the class notes. Which bet is riskiest and which best is safest?

a) You get $5 with probability 1.0.
b) You get $10 with probability 0.5, or $0 with probability 0.5.
c) You get $5 with probability 0.5, $10 with probability 0.25 and $0 with probability 0.25.
d) You get $5 with probability 0.5, $105 with probability 0.25 or lose $95 with probability 0.25.

Question 8) Let X be a random variable with E(X) = 20 and Var(X) = 10. Find the following.

a) E ( X² )
b) E ( 3X + 10 )
c) E (-X)
d) Standard deviation of -2X ?

Binomial Problems [feel free to use R when possible]

Question 9) The random variable X has a binomial distribution with E(X)=18 and Var(X)=7.2. Find n and pfor this distribution [use the formulas for mean and variance].

Question 10) Suppose X is a binomial random variable with n=15 and p=0.3. Feel free to use a computer to answer the following:

a) P(X=0)
b) P(X=2)
c) P(X<2)
d) P(X>8)
e) E(X)
f) Var(X)

Question 11) A school newspaper reporter decides to randomly survey 12 students to see if they will attend YardFest TM festivities this year. Based on past years, she knows that 22% of students attend YardFest TM festivities. We are interested in the number of students who will attend the festivities.

a) How many of the 12 students do we expect to attend the festivities?
b) Find the probability that at most 4 students will attend.
c) Find the probability that more than 2 students will attend.

Question 12) Suppose that about 90% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen.

a) How many are expected to attend their graduation?
b) Find the probability that 17 or 18 attend.
c) Based on numerical values, would you be surprised if all 22 attended graduation? Justify your answer numerically.

Question 13) According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 75% of the time. Suppose 15 flights are randomly selected, and the numberof on-time flights is recorded.

a) Explain why this is a binomial experiment.
b) Find and interpret the probability that exactly 10 flights areon time.
c) Find and interpret the probability that between 8 and 10flights, inclusive, are on time.
d) Find the mean and variance of the number of on time flights.

Continuous Random Variables[feel free to use R when possible]

Question 14) Let X be a uniformly distributed random variable on the interval 0 to 1

a) Calculate P(X = 0.25)
b) Calculate P(0.7 c) Calculate the expected value of X
d) Calculate E(X²)

Question 15) For the standard normal random variable Z, compute the following:

a) P(0 ≤ Z ≤ 0.73)
b) P(-1.50 ≤ Z ≤ 0)
c) P(Z ≥ .0.44)
d) P(-1.50 ≤  Z ≤ 0.40)
e) P(Z ≤ 5.23)
f) E(3- 4Z)
g) Var(4 -3Z)

Question 16) The time needed to hand stitch a Swoosh soccer ball isnormally distributed with mean 43minutes and standard deviation 3.1 minutes. If 3 workers start at the same time, what is the probability that at least one of them will complete their soccer ball in under 44 minutes?

Question 17) The weight of reports produced in a certain department has a Normal distribution with mean 60g and standard deviation 12g. What is the probability that the next report will weigh less than 45g?

Question 18) The times taken to complete an introduction to business statistics exam have a normal distribution with a mean of 65 minutes and standard deviation of 7 minutes. There are 150 students who took the exam and students are allowed a total of 75 minutes to take the exam.

a) What is the chance that Mike finished his exam in 63 to 72 minutes?
b) What is the expected number of students who finished in less than 75 minutes?
c) As some students were not able to finish the exam in time, the instructor allowed 6 more minutes. Given he already spent 75 minutes on the exam, what is the chance that Chris finished his exam in extended time, that is between 75 and 81 minutes

Question 19) Suppose X is a continuous random variable taking values between 0 and 2 and having the probability density function below. Calculate P(1 ≤ X ≤ 2)

Basic Probability Problems

Question 20) A die is ‘fixed' so that certain numbers will appear more often. The probability that a 6 appears is twice the probability of a 5 and 3 times the probability of a 4. The probabilities of 3, 2 and 1 are unchanged from a normal die. The probability distribution table is given below.

x	1	2	3	4	5	6
Pr(x)	1/6	1/6	1/6	x/3	x/2	x

a) What is the value of x in the probability distribution ?
b) What is the probability of getting a ‘double' with two of these dice. Compare with the ‘normal' probability of getting a double when the dice is fair.

Question 21) Find the missing profit (or loss) so that the following probability table has an expected value of 0.

x	4	5	6	7	8	9	10
Pr(X = x)	0.1	0.06	025	0.16	0.09	0.21	0.13
Gain	-3	4	-2	5	-8	12