Exam 2 Stat 4710

What is the definition of the probability density function for a continuous random variable Y?

Let F(y) be the distribution function for a continuous random variable Y. Then f (y), given by𝑓 (𝑦) = 𝑑𝐹(𝑦)/ 𝑑𝑦 = 𝐹'(𝑦) wherever the derivative exists, is called the probability density function for the random variable Y PDF is f(y)

What is the random variable Y for the poisson distribution for a discrete random variable?

Let Y denote the number of occurrences in an interval of length 1

For the continuous random variable Y. P(Y=a)=____

P(Y=a)=0 where a is a constant

What is the definition of the distribution function of any random variable Y?

The distribution function of any random variable Y, denoted by F(y), is such that 𝐹(𝑦) = 𝑃(𝒀 ≤ 𝒚) for −∞< y < ∞

What is the random variable Y for the uniform distribution

The random variable Y has a uniform distribution if its pdf is equal to a constant on its support

What is the moment generating function of the poisson distribution?

*using Taylor series

Let Y be a random variable with a mean of 19 and a variance of 9. Use Tchebysheff's theorem to find the following. (a)a lower bound for P(15 < Y < 23) (b)the value of C such that P(|Y − 19| ≥ C) ≤ 0.09

3.11 #1a&b

Ninety percent of the engines manufactured on an assembly line are defective. Suppose engines are randomly selected one at a time and tested. (Round your answers to three decimal places.) (a)What is the probability that the third nondefective engine will be found on the fifth trial? (b)What is the probability that the third nondefective engine will be found on or before the fifth trial?

3.4 #4

The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send four employees who have positive indications of asbestos to a medical center for further testing. If 40% of the employees have positive indications of asbestos in their lungs, find the probability that fifteen employees must be tested in order to find four positives. (Round your answer to three decimal places.)

3.6 #1

The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send two employees who have positive indications of asbestos to a medical center for further testing. If 40% of the employees have positive indications of asbestos in their lungs, find the probability that fifteen employees must be tested in order to find two positives. If each test costs $40, find the expected value in dollars and variance of the total cost of conducting the tests necessary to locate the two positives.

3.6 #2

The random variable Y has a Poisson distribution and is such that p(0) = p(1). What is p(4)? (Round your answer to three decimal places.)

3.8 #1

Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of eight per hour. If it takes approximately ten minutes to serve each customer, find the mean and variance of the total service time in minutes for customers arriving during a 1-hour period. (Assume that a sufficient number of servers are available so that no customer must wait for service.) Is it likely that the total service time will exceed 3.5 hours? (Round your answer to three decimal places.) Since P(service time exceeds 3.5 hours) = _____, this is an unlikely or likely event.

3.8 #2

Let Y denote a random variable that has a Poisson distribution with mean 𝜆 = 3. (Round your answers to three decimal places.) (a)Find P(Y = 7). (b)Find P(Y ≥ 7). (c)Find P(Y < 7). (d)Find P(Y ≥ 7|Y ≥ 3).

3.8 #3

Cars arrive at a toll booth according to a Poisson process with mean 70 cars per hour. Suppose the attendant makes a phone call. How long, in seconds, can the attendant's phone call last if the probability is at least 0.6 that no cars arrive during the call? (Round your answer to the nearest integer.)

3.8 #4

The number of knots in a particular type of wood has a Poisson distribution with an average of 1.6 knots in 10 cubic feet of the wood. Find the probability that a 10-cubic-foot block of the wood has at most 1 knot. (Round your answer to three decimal places.)

3.8 #5

A salesperson has found that the probability of a sale on a single contact is approximately 0.01. If the salesperson contacts 600 prospects, what is the approximate probability of making at least one sale? (Round your answer to three decimal places.)

3.8 #6

The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Assume that the tunnel is observed during eleven two-minute intervals, thus giving eleven independent observations Y1, Y2, , Y11, on the Poisson random variable. Find the probability that Y > 5 during at least one of the eleven two-minute intervals. (Round your answer to three decimal places.)

3.8 #7

In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean 𝜆 = 3. The profit per foot when the rope is sold is given by X, where X = 80 − 4Y − Y2. Find the expected profit per foot.

3.8 #8

A food manufacturer uses an extruder (a machine that produces bite-size cookies and snack food) that yields revenue for the firm at the rate of $200 per hour when in operation. However, the extruder breaks down on an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is R = 1,900 − 50Y2. Find the expected daily revenue in dollars for the extruder.

3.8 #9

If Y has a binomial distribution with trials and probability of success p, show that the moment-generating function for Y is m(t)=(pe^t + q)^n, where q=1-p

3.9 #145

If Y has a geometric distribution with a probability of success p, show that the moment-generating function for Y is m(t)=(pe^t)/(1-qe^t) where q=1-p

3.9 #147

Differentiate the moment-generating function in Exercise 3.147 to find E(Y ) and E(Y^2). Then find V(Y ). m(t)=(pe^t)/(1-qe^t) where q=1-p

3.9 #148

Refer to m(t)=(pe^t + q)^n, where q=1-p. Use the uniqueness of moment-generating functions to give the distribution of a random variable with moment-generating function m(t)=(.6e^t + .4)^3

3.9 #149

Refer tom(t)=(pe^t)/(1-qe^t) where q=1-p. Use the uniqueness of moment-generating functions to give the distribution of a random variable with moment-generating function m(t)= (.3e^t)/(1-.7e^t)

3.9 150

Refer to m(t)=(pe^t + q)^n, where q=1-p. If Y has moment-generating function , m(t)=(.7e^t + .3)^10 what is P(Y<=5) ?

3.9 151

Refer to Example 3.23. If Y has moment-generating function m(t)=e^6(e^t−1) , what is P(|Y − µ| ≤ 2σ)?

3.9 152

Find the distributions of the random variables that have each of the following moment-generating functions: a. m(t)=[(1/3)e^t +(2/3)]^5 b .m(t)=e^t/(2−e^t). c . m(t)=e^2(e^t -1)

3.9 153

Let m(t)=(1/6)e^t+(2/6)e^2t+(3/6)e^3t. Find the following: a. E(Y ) b. V(Y ) c. The distribution of Y

3.9 155

Suppose that Y is a random variable with moment-generating function m(t). a What is m(0)? b If W = 3Y, show that the moment-generating function of W is m(3t). c If X = Y − 2, show that the moment-generating function of X is e^−2tm(t).

3.9 156

Refer to Exercise 3.156. a If W = 3Y, use the moment-generating function of W to show that E(W) = 3E(Y ) and V(W) = 9V(Y ). b If X = Y − 2, use the moment-generating function of X to show that E(X) = E(Y ) − 2 and V(X) = V(Y ).

3.9 157

A random variable Y has the following distribution function. F(y) = P(Y ≤ y) = 0, for y < 2 1/8,for 2 ≤ y < 2.5 3/16,for 2.5 ≤ y < 4 1/2,for 4 ≤ y < 5.5 9/16,for 5.5 ≤ y < 6 11/16,for 6 ≤ y < 7 1,for y ≥ 7 (a)Is Y a continuous or discrete random variable? Why? OPTION 1. Y is discrete because F(y) is a continuous function and the set of possible values of Y is an uncountable set. 2. Y is continuous because F(y) is a continuous function and the set of possible values of Y is an uncountable set. 3. Y is discrete because F(y) is not a continuous function and the set of possible values of Y is a countable set. 4. Y is continuous because F(y) is not a continuous function and the set of possible values of Y is a countable set.

4.2 #4a Y is discrete because F(y) is not a continuous function and the set of possible values of Y is a countable set.

A random variable Y has the following distribution function. F(y) = P(Y ≤ y) = 0, for y < 2 1/8,for 2 ≤ y < 2.5 3/16,for 2.5 ≤ y < 4 1/2,for 4 ≤ y < 5.5 9/16,for 5.5 ≤ y < 6 11/16,for 6 ≤ y < 7 1,for y ≥ 7 What values of Y are assigned positive probabilities? (Enter your answers as a comma-separated list.)

4.2 #4b

A random variable Y has the following distribution function. F(y) = P(Y ≤ y) = 0, for y < 2 1/8,for 2 ≤ y < 2.5 3/16,for 2.5 ≤ y < 4 1/2,for 4 ≤ y < 5.5 9/16,for 5.5 ≤ y < 6 11/16,for 6 ≤ y < 7 1,for y ≥ 7 What is the median, 𝜑0.5, of Y ?

4.2 #4d

Suppose that Y possesses the density function f(y) = cy, 0 ≤ y ≤ 2, 0, elsewhere. Find the value of c that makes f(y) a probability density function

4.2 #5a

Suppose that Y possesses the density function f(y) = cy, 0 ≤ y ≤ 2, 0, elsewhere. Find F(y)

4.2 #5b

Suppose that Y possesses the density function f(y) = cy, 0 ≤ y ≤ 2, 0, elsewhere. graph f(y) and F(y)

4.2 #5c

Suppose that Y possesses the density function f(y) = cy, 0 ≤ y ≤ 2, 0, elsewhere. Use F(y) to find P(1 ≤ Y ≤ 2).

4.2 #5d

Suppose that Y possesses the density function f(y) = cy, 0 ≤ y ≤ 2, 0, elsewhere. Use f(y) and geometry to find P(1 ≤ Y ≤ 2).

4.2 #5e

A supplier of kerosene has a 150-gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If Ydenotes weekly demand in hundreds of gallons, the relative frequency of demand can be modeled by f(y) = y, 0 ≤ y ≤ 1, 1, 1 < y ≤ 1.5, 0, elsewhere. Find F(y)

4.2 #6a

A supplier of kerosene has a 150-gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If Ydenotes weekly demand in hundreds of gallons, the relative frequency of demand can be modeled by f(y) = y, 0 ≤ y ≤ 1, 1, 1 < y ≤ 1.5, 0, elsewhere. Find P(0 ≤ Y ≤ 0.3).

4.2 #6b

A supplier of kerosene has a 150-gallon tank that is filled at the beginning of each week. His weekly demand shows a relative frequency behavior that increases steadily up to 100 gallons and then levels off between 100 and 150 gallons. If Ydenotes weekly demand in hundreds of gallons, the relative frequency of demand can be modeled by f(y) = y, 0 ≤ y ≤ 1, 1, 1 < y ≤ 1.5, 0, elsewhere. Find P(0.3 ≤ Y ≤ 1.3).

4.2 #6c

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1, 0,elsewhere. Find c.

4.2 #8a

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1,0,elsewhere. Find F(y)

4.2 #8b

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1, 0,elsewhere. Use F(y) in part (b) to find F(-1), F(0), and F(1)

4.2 #8d

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1, 0,elsewhere. Find the probability that a randomly selected student will finish in less than three-quarters of an hour. (Round your answer to four decimal places.)

4.2 #8e

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1, 0,elsewhere. Given that a particular student needs at least 30 minutes to complete the exam, find the probability that she will require at least 45 minutes to finish. (Round your answer to four decimal places.)

4.2 #8f

Let the distribution function of a random variable Y be F(y) = 0, y ≤ 0, y/8, 0 < y < 2, y^2/16, 2 ≤ y < 4, 1,y ≥ 4. Find the density function of Y.

4.2 #9a

Let the distribution function of a random variable Y be F(y) = 0, y ≤ 0, y/8, 0 < y < 2, y^2/16, 2 ≤ y < 4, 1,y ≥ 4. Find P(1.50 ≤ Y ≤ 3.25). (Round your answer to four decimal places.)

4.2 #9b

Let the distribution function of a random variable Y be F(y) = 0, y ≤ 0, y/8, 0 < y < 2, y^2/16, 2 ≤ y < 4, 1,y ≥ 4. Find P(Y ≥ 2.00). (Round your answer to four decimal places.)

4.2 #9c

Let the distribution function of a random variable Y be F(y) = 0, y ≤ 0, y/8, 0 < y < 2, y^2/16, 2 ≤ y < 4, 1,y ≥ 4. Find P(Y ≥ 1.50|Y ≤ 3.25).

4.2 #9d

The length of time required by students to complete a one-hour exam is a random variable with a density function given by f(y) = cy^2 + y, 0 ≤ y ≤ 1, 0,elsewhere. Graph f(y) and F(y)

4.2 8c

A random variable Y has the following distribution function. F(y) = P(Y ≤ y) = 0, for y < 2 1/8,for 2 ≤ y < 2.5 3/16,for 2.5 ≤ y < 4 1/2,for 4 ≤ y < 5.5 9/16,for 5.5 ≤ y < 6 11/16,for 6 ≤ y < 7 1,for y ≥ 7 Find the probability function for Y. (Order your answers from smallest to largest, first by y, then by p(Y = y). For each answer, enter an exact number as an integer, fraction, or decimal.) p(Y =__ )=__ p(Y =__) =__ p(Y =__) =__ p(Y =__ )=__ p(Y =__)=__ p(Y =__)=__

4.32# 4c

A random variable Y has a uniform distribution over the interval (𝜃1, 𝜃2). Derive the variance of Y. Find E(Y)^2 in terms of (𝜃1, 𝜃2). E(Y)^2 = ___________ Find E(Y2) in terms of (𝜃1, 𝜃2). E(Y^2) =___________ Find V(Y) in terms of (𝜃1, 𝜃2). V(Y) = ______________

4.4 #1

A circle of radius r has area A = 𝜋r2. If a random circle has a radius that is uniformly distributed on the interval (2, 3),what are the mean and variance of the area of the circle?

4.4 #2

Upon studying low bids for shipping contracts, a microcomputer manufacturing company finds that intrastate contracts have low bids that are uniformly distributed between 20 and 26, in units of thousands of dollars. (a)Find the probability that the low bid on the next intrastate shipping contract is below $22,000. (Round your answer to four decimal places.) (b)Find the probability that the low bid on the next intrastate shipping contract is in excess of $24,000. (Round your answer to four decimal places.)

4.4 #3

The weekly amount of money spent on maintenance and repairs by a company was observed, over a long period of time, to be approximately normally distributed with mean $395 and standard deviation $25. Suppose that $455 is budgeted for next week; use R to calculate the probability that the actual costs will exceed the budgeted amount. (Round your answer to five decimal places.)

4.5 #1 0.008197536

The grade point averages (GPAs) of a large population of college students are approximately normally distributed with mean 2.6 and standard deviation 0.7. If students possessing a GPA less than 1.75 are dropped from college, what percentage of the students will be dropped? (Round your answer to two decimal places.)

4.5 #2 11.23193%

Suppose the number of customers arrive at a bakery follows a poisson process at an average rate of 15 per hour. What is the probability that it takes less than/ more than 10 minutes for the first 3 customers to arrive? What is the average amount of the time that will elapse before 3 arrivers?

4.6

What is the random variable Y for the gamma distribution?

the waiting time Y until the αth occurrence in an approximate Poisson process (λ), has a gamma distribution with parameters α and Beta= 1/λ.

What are the 3 assumptions for the poisson probability distribution?

a. Events are independent of each other in nonoverlapping subintervals. b. The average rate (the number of events per unit) is constant. c. Two events cannot occur at the same time).

What is the random variable Y for the negative binomial distribution for a discrete random variable?

distributedY indicated the number of trials needed to observe the rth success, then Y is negative binomial distributed.

Suppose that 𝑌 has a binomial distribution with 𝑛 = 2 and 𝑝 = 1/2. Find 𝐹(𝑦).

Ex from 4.2

What is the relationship between CDF and PDF for the continuous variables?

F'(Y)=f(Y) or ∫f(t) dt from −∞ to ∞

What is the definition of the cumulative density function fr a continuous random variable Y?

F(y) accumulates all of the probability less than or equal to x. The cumulative distribution function for continuous random variables is just a straightforward extension of that of the discrete case. All we need to do is replace the summation with an integral.

What are the properties of a CDF for a continuous random variable?

F(y)is a distribution function, then1). 1). F(−∞) = lim y→−∞F(y) = 0 2). F(∞) = lim y→∞F(y) =1 3). F(y) is a nondecreasing function of y. [If y1 and y2 are any values such that y1 < y2, then 𝐹(𝑦1) ≤ 𝐹(𝑦2).] 3. simply put F(y)= 𝑃(𝒀 ≤ 𝒚)

For the continuous random variable Y. 𝑃(𝑎 ≤ 𝑌 < 𝑏)=___________=__________=__________

𝑃(𝑎 ≤ 𝑌 < 𝑏)= 𝑃(𝑎 ≤ 𝑌 ≤ 𝑏)= 𝑃(𝑎 < 𝑌 ≤ 𝑏)= 𝑃(𝑎 < 𝑌 < 𝑏)