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A random variable X has its probability function given by 1 T2 T f(x) 0.30 0.10 C L 0.2 0.4c a) [3.5 marks] Find c and F(x), the cumulative distribution function for X (for all real values of X).

Problem 3 Consider the two independent random variables X1 and X2, such that E[X = Hi, i = 1,2 and Var(X;) = o?, i = 1,2. Let Y 2X1-3X2. Find the mean and variance of Y in terms of the means and variances of X1 and X2. %3D

QUESTION (2): (Marks: 6,4) (a) If X and Y are independent binomial random variables with identical parameters (n, e), show that P( X = K | X+Y = m)= (b) Let random variable X has binomial distribution with parameters (n = 100, 0 = 0.2). Use normal approximation to find P( X > 25 ).

Let X, and X, be independent random variables, X, has pdf f(x) = 6x(1 – x) for 0 < x < 1, and X2 has pmf f(x) = x/17 for x = 3,5,9. a) Find the mean and variance of 7X1 - 3X2. b) Find E(XXX).

1. As part of their customer service assessment, a manufacturer compiled the table below showing wait times (rounded to the nearest minute) for calls to their appliance help center. Wait time (min) 1 2 3 4 5 6 7 8 9 10 Number 16 55 87 101 87 63 42 20 16 13 a. What is P(X = 1)? b. What is P(X= 5)? c. What is P(X = 10)? d. What is the value of the cumulative distribution function for this random variable in the range 1 sx< 2? e. What is the value of the cumulative distribution function for this random variable in the range 5 SX < 6? f. What is the value of the cumulative distribution function for this random variable in the range 9 S x < 10? g. What is the mean of the random variable in this problem? h. What is the variance for the random variable in this problem? i. Assume that five people independently place a service call to the appliance help center What is the probability that the waiting time of exactly one person is less than or equal to 4 min? j. Assume that five people independently place a service call to the appliance help center. What is the probability that exactly two people wait more than 4 min? k. Assume that five people independently place a service call to the appliance help center. What is the probability that at least one person waits more than 4 min?

to be Problem 1 In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the probability that a wafer passes the test is 0.7 and that wafers are independent. (a) Determine the probability mass function of the number of wafers from a lot that pass the test. (b) Determine the cumulative distribution function for the random variable (c) Determine the mean and variance of the random variable

Question 5 If the range of X is the set {0,1,2,3,4,5,6,7,8} and P(X = x) is defined in the following table: x 0 1 3 4 5 6 7 8 PCX = x) 0.1965 0.2155 0.1898 0.0539 0.01785 0.0539 0.08435 0.08089 0.1078 determine the mean and variance of the random variable. Round your answers to two decimal places. (a) Mean = (b) Variance = Determine the mean and variance of the random variable with the following probability mass function. f(x) = (27/13)(1/3)*, x = 1,2,3 Round your answers to three decimal places (e.g. 98.765). Mean = Variance = The following function is probability mass function. 2x +5 f(x) = 1 X=0,1,2,3,4 45 Determine the mean, H, and variance, o2, of the random variable. Round your answers to two decimal places (e.g. 98.76). = O2 =

Problem A: Let X1 and X2 be independent random variables, X1 has pdf f (x) = 6x(1 – x) for 0 < x < 1, and X2 has pmff (x) = x/17 for x = 3,5,9. a) Find the mean and variance of 7X1 - 3X2. b) Find E (X{X²). Problem B: Let X, and X2 be independent random variables, X, is binomial with parameters 4 and 0.7, and X2 is binomial with parameters 5 and 0.5. a) Find P(X,X, = 8). b) Find P(X, + X2 = 8).

Let the random variable X have a discrete uniform distribution on the integers 8 sxs 15. Determine the mean, w, and variance, o, of X. Round your answers to two decimal places (e.g. 98.76). J = o2= Assume that the wavelengths of photosynthetically active radiations (PAR) are uniformly distributed at integer nanometers in the red spectrum from 620 to 655 nm. What is the mean and variance of the wavelength distribution for this radiation? Round your answers to one decimal place (e.g. 98.7). Mean = Variance =

3. For this question let p = X/n, where X is a Binomial(n, p) random variable the number of successes in n independent trials, each with success probability p. i. Derive the variance of p as a function of n and p. Your proof should not rely on existing formulas for the variances of sample proportions or Binomial random variables, but it can take for granted that for independent random variables {V;}, Var(E; V:) = E, Var(V;), and that for constant a and random V, Var(aV) = a²Var(V). (Hint: Recall that a Binomial(n, p) random variable is the sum of n independent Bernoulli(p) random variables.) ?.

Discrete-Event System Simulation |(5th Edition) See this solution in the app Chapter 5, Problem 13E Bookmark Show all steps: OON Problem A random variable X that has pmf given by p(x)= 1/(n + 1) over the range Rx = {0, 1, 2, ..., n} is said to have a discrete uniform distribution. (a) Find the mean and variance of this distribution. Hint: = 'on 4 1 and = nou + 1}{2n + 1) (b) If Rx = {a, a + 1, a + 2, ...,b}, compute the mean and variance of X.

Problem 1: 15 points (= 5 + 5 + 5] You are given that an integer-valued random variable N has distribution such that P[N 2 n] = (%)' for n 20. Non-negative random variables X = {Xk: k = 1, 2, ...,} do not depend on N and share the common distribution with density f*(x) = (216) - 23.-6x valid for 2 > 0 Consider a random sum, Xk. k=1 1. Derive expectation of S, that is: ES 2N+5 2. Evaluate variance of S, that is: Var (S] 3. Find the marginal second moment of S, that is: E(S)

Problem 2: 15 points (= 5 + 5 + 5] Assume that an integer-valued random variable N has distribution P[N = n] = - valid for n > 0. Once N = n is observed, a variable X is defined as the number of successes after M = (2N+4) independent binary trials with success rate q = 0.6, or equivalently, for integer k, such that 0 Sk <2N + 4, the conditional distribution of X is: (2N+4)! P ( X = k N = n] = 1 1. Find expected value of X, that is E[X] 2. Derive variance of X, that is: Var [X] 3. Determine the marginal second moment of X, that is EX) k!. (2N +4 1-(0.6)* . (0.4)2N+4-6

Suppose that the height of Asian males aged 35–44 in the United States is normally distributed with a standard deviation of 3 in. Jeremy takes a simple random sample of 16 such men and notices that none of them are unusually tall or unusually short. He then calculates the average of the sample to be 67 in. and is planning on using this to find a 99% z-confidence interval for the true mean height. Can Jeremy use this information to find the z-confidence interval? Complete the following sentences. the sample The population distribution is and the population standard deviation is to find a 99% z-confidence interval. Therefore,

Problem 1: 15 points (= 5 + 5 + 5) You are given that an integer-valued random variable N has distribution such that P[N 2 n] = ()' for n 20. Non-negative random variables X = {Xx : k = 1, 2, ...,} do not depend on N and share the common distribution with density f*(x) = (216). 23.-6 valid for x > 0 Consider a random sum, 2N+5 S= XR. k1 1. Derive expectation of S, that is: ES] 2. Evaluate variance of S, that is: Var (S) 3. Find the marginal second moment of S, that is: E(S)

Problem 1: 15 points (= 5 + 5 + 5] Assume that a discrete random variable (N) is geometrically distributed, with P[N 2 n] = (-1)' for n = 0, 1, .... An n OT Given N = n, a random variable (Y) has conditional distribution with density, pYIN (y\n) = .3.'.ev for y>0. 1. Determine the marginal density of Y. 2. Show that for any natural r, the moment EY'] is finite and evaluate it. 3. Derive variance of Y, that is Var [Y] = E [Y2) – (E [Y])?

Question 5 of 18 Suppose that the height of Asian males aged 35-44 in the United States is normally distributed with a standard deviation of 3 in. Jeremy takes a simple random sample of 19 such men and notices that none of them are unusually tall or unusually short. He then calculates the average of the sample to be 67 in, and is planning on using this to find a 90% z-confidence interval for the true mean height. Can Jeremy use this information to find the z-confidence interval? Complete the following sentences. The population distribution is the sample and the population standard deviation is Therefore, to find a 90% z-confidence interval.

Value at Risk Safe Bank (SB) calculates “Value at Risk” at 90% confidence level monthly, SB estimates mean and standard deviation of monthly returns on its portfolio to be 2% and 6%, respectively. Current portfolio value is $150 million a. If returns are normally distributed, what currently is the VAR of SB?

Problem 2: 15 points [= 5 + 5 + 5] Assume that a random variable (Y) follows the exponential distribution with density fY (y) = 2e-2y for y >0 and fY (y) = 0 elsewhere. Given Y = y, a random variable N is Poisson distributed with rate = y. 1. Determine marginal distribution of N. 2. Evaluate expectation of N. 3. Derive variance of N.

Problem 1: 15 points [= 5 + 5 + 5] Assume that a discrete random variable (N) is geometrically distributed, with P[N > n] = () for n = 0, 1, .... %3D Given N = n, a random variable (Y) has conditional distribution with density, fYIN (y|n) = • y'. e- for y > 0. n! 1. Determine the marginal density of Y. 2. Show that for any natural r, the moment E(Y'] is finite and evaluate it. 3. Derive variance of Y, that is Var (Y] = E (Y) – (E[Y])²

It is known that the expected monthly return of stock ABS is 2%. Also the monthly standard deviation is estimated to be 6%. You can assume that the monthly returns are normally distributed. a) compute the probability that any given months return will exceed 4% b) compute the probability that the average monthly return is any given year exceed 4% c) Design a 99% confidence interval for the monthly return of the stock. Also use the relevant graph to support your answer.

Consider the following frequency distribution. Class 2 up to 4 Frequency 20 4 up to 6 6 up to 8 8 up to 10 60 80 20 a. Calculate the population mean. (Round your answer to 2 decimal places.) Population mean b. Calculate the population variance and the population standard deviation. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.) Population variance Population standard deviation

A random variable Xfollows the continuous uniform distribution with a lower bound of -5 and an upper bound of 7. a. What is the height of the density function fix? (Round your answer to 4 decimal places.) points eBoos b. What are the mean and the standard deviation for the distribution? (Round your answers to 2 decimal places.) References Mean Standard deviation c. Calculate XS-1). (Round Intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.) PAX 5.1) The scheduled arrival time for a daily flight from Boston to New York is 9:25 am. Historical data show that the arrival time follows the continuous uniform distribution with an early arrival time of 9:16 am and a late arrival time of 9:49 am a. After converting the time data to a minute scale, calculate the mean and the standard deviation of the distribution. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) points minutes Mean Standard deviation minutes 989 b. What is the probability that a flight arrives late (later than 9:25 am)? (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) References Probability ter 6 Homework Assignment Find the following probabilities based on the standard normal variable 2 (You may find it useful to reference the z table. Round your answers to 4 decimal places.) a. P(Z >0.75) b. PIZS-24) C POSZ $1.31) d. P(-0.56 $2$2.59) ?19

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 475 stocks. Let be a random variable that represents the monthly percentage return for this fund. Suppose has mean = 1.6% and standard deviation = 1.2%. (a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that (the average monthly return on the 475 stocks in the fund) has a distribution that is approximately normal? Explain. ----- Yes , is a mean of a sample of = 475 stocks. By the  --central limit theory--- central limit theorem law of large numbers , the distribution  --is not approximately normal. (b) After 9 months, what is the probability that the monthly percentage return will be between 1% and 2%? (Round your answer to four decimal places.)_______________ (c) After 18 months, what is the probability that the monthly percentage return will be between 1% and 2%? (Round your answer to four decimal places.)________________ (d) Compare your answers to parts (b) and (c). Did the probability increase as (number of months) increased? Why would this happen? (e) If after 18 months the average monthly percentage return is more than 2%, would that tend to shake your confidence in the statement that = 1.6%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.) ( > 2%) = _________________ Explain. This is very unlikely if = 1.6%. One would suspect that the European stock market may be heating up. This is very likely if = 1.6%. One would not suspect that the European stock market may be heating up.     This is very likely if = 1.6%. One would suspect that the European stock market may be heating up. This is very unlikely if = 1.6%. One would not suspect that the European stock market may be heating up. 12) The taxi and takeoff time for commercial jets is a random variable with a mean of 8.5 minutes and a standard deviation of 2.5 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway. (a) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)_________ (b) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)_________ (c) What is the probability that for 33 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)______________

Problem 2: 15 points (= 5 + 5 + 5] Assume that a random variable (Y) follows the exponential distribution with density f(y) = 2e-2 for y>0 and f (y) = 0 elsewhere. Given Y = y, a random variable N is Poisson distributed with rate = y. 1. Determine marginal distribution of N. 2. Evaluate expectation of N. 3. Derive variance of N.

The Downtown Parking Authority of Tampa, Florida, reported the following information for a sample of 382 customers on the number of hours cars are parked and the amount they are charged. Number of Hours Frequency 41 35 WN Amount Charged $ 3.00 6.00 8.00 9.00 11.00 12.00 16.00 18.00 00 O 382 (a-1) Convert the information on the number of hours parked to a probability distribution. (Round your answers to 3 decimal places.) Hours Probability 1 .107 0 .092 .147 0 2 .157 .131 0 0 .128 8 .118 0 .120 (a-2) Is this a discrete or a continuous probability distribution? Discrete Continuous (b-1) Find the mean and the standard deviation of the number of hours parked. (Round your Mean to 3 decimal places and the standard deviation to 3 decimal places.) Mean 4.569 Standard deviation 2.186 (b-2) How long is a typical customer parked? (Round your answer to 3 decimal places.) The typical customer is parked for hours parked for hours ] (c) Find the mean and the standard deviation of the amount charged. (Round your answers to 2 decimal places.) 10.49 Mean Standard deviation

The polo Development Firm is building a shopping center. It has informed renters that their rental 10. spaces will be ready for occupancy in 18 months. If the expected time until the shopping center is completed is estimated to be 15 months, with a standard deviation of 5 months, what is the probability that the renters will not be able to occupy in 18 months? Distribution Standard deviation Probability NOT occupied Q10 Mean Occupied Normal Occupuied 15 5 18 Not occupied

Assume a normal distribution and find the following probabilities. ( < 17 | = 21 and = 3) enter the probability of fewer than 17 outcomes if the mean is 21 and the standard deviation is 3 ( ≥ 76 | = 60 and = 9) enter the probability of 76 or more outcomes if the mean is 60 and the standard deviation is 9 ( > 56 | = 60 and = 5) enter the probability of more than 56 outcomes if the mean is 60 and the standard deviation is 5 (17 < < 20 | = 19 and = 3) enter the probability of more than 17 and fewer than 20 outcomes if the mean is 19 and the standard deviation is 3 ( ≥ 94 | = 80 and = 2.88) enter the probability of 94 or more outcomes if the mean is 80 and the standard deviation is 2.88

The polo Development Firm is building a shopping center. It has informed renters that their rental 10. spaces will be ready for occupancy in 18 months. If the expected time until the shopping center is completed is estimated to be 15 months, with a standard deviation of 5 months, what is the probability that the renters will not be able to occupy in 18 months? Distribution Standard deviation Probability NOT occupied Q10 Mean Occupied Normal Occupuied 15 5 18 Not occupied

Suppose random variables Z, are exponentially distributed: Zi Exp(2) for i = 1,2,...,n and assume that the random variables 2; are independent. For the definition and properties of the exponential distribution, refer to the formula sheet from the second lecture. For each of the following functions of the Zi, find the expectation E[ ] and variance Var[ ]. (a) 3 21 -5 (b) 1.5 Z1 +2 22 - 3 (c) 1-32 (simplify as much as possible, but final answer will contain )

A study of the time spent shopping in a supermarket for 20 specific items showed an approximately uniform distribution between 20 minutes and 30 minutes. a. What is the probability that the shopping time will be between 24 and 26 minutes? b. What is the probability that the shopping time will be less than 27 minutes? c. What are the mean and standard deviation of the shopping time? a. The probability that the shopping time will be between 24 and 26 minutes is (Type an integer or a decimal.) . b. The probability that the shopping time will be less than 27 minutes is (Type an integer or a decimal.) c. The mean of the given uniform distribution is u=17 (Type an integer or a decimal.) The standard deviation of the given uniform distribution is o = (Round to four decimal places as needed.)

1. Probability density function of exponential random variable and general form of the characteristic function described as below. fx(x) = 10 x < 0 $,(W) = [e/x] = 1 el fx(x) dx In addition to these equations moments of X random variable can be obtained from the characteristic function E[X] = 00:(W) 1 dwo a) Find the characteristic function for an exponentially distributed random variable. b) Find the variance of exponential random variable from characteristic function of exponentially distributed random variable.

Customer Satisfaction Ratings Number of Customers Rating Restaurant A Restaurant B 5 17 13 15 AWN 13 14 UP Print Done The owner of two fast-food restaurants has recorded customer satisfaction ratings for both locations on a scale of 1 to 515 - Most satisfied). The table linked below summarizes the data a. Calculate the mean satisfaction rating at each location b. Calculate the standard deviation of each distribution c. What conclusions can be drawn from these results? Click the icon to view the customer satisfaction ratings a. What is the mean for Restaurant A? (Type an integer or decimal rounded to three decimal places as needed) What is the mean for Restaurant B? (Type an integer or decimal rounded to three decimal places as needed.) b. What is the standard deviation for Restaurant A? GE (Type an integer or decimal rounded to three decimal places as needed.) What is the standard deviation for Restaurant B? = (Type an integer or decimal rounded to three decimal places as needed.) c. What conclusions can be drawn from these results? Consider the discrete probability distribution below. Complete parts a and b to the right. Outcome Probability 0.24 0.27 0.19 0.12 0.09 0.06 0.03 a. Calculate the mean of this distribution - (Type an integer or a decimal.) b. Calculate the standard deviation of this distribution ܕ ܐ ܕ G (Round to three decimal places as needed.) ܗ Question Help A political committee consists of five Democrats and seven Republicans. A subcommittee of six people needs to be formed from this group. For this problem, define a success as a Democrat being selected for the subcommittee) a. Determine the probability that this subcommittee will consist of four Democrats and two Republicans were randomly selected b. Calculate the mean and standard deviation of this distribution a. The probability that this subcommittee will consist of four Democrats and two Republicans if they were randomly selected (Round to four decimal places as needed.) b. The mean of this distribution is (Round to three decimal places as needed) The standard deviation of this distribution is (Round to three decimal places as needed) Consider the discrete probability distribution below. Complete parts a and b to the right a. Calculate the mean of this distribution Outcome وه ش به ا Probability 0.24 0 27 0.19 0.12 0.09 0.06 0.03 (Type an integer or a decimal.) b. Calculate the standard deviation of this distribution. ه ن (Round to three decimal places as needed) و Question Help A political committee consists of five Democrats and seven Republicans. A subcommittee of six people needs to be formed from this group. For this problem, define a success as a Democrat being selected for the Subcommittee) a. Determine the probability that this subcommittee will consist of four Democrats and two Republicans if they were randomly selected b. Calculate the mean and standard deviation of this distribution a. The probability that this subcommittee will consist of four Democrats and two Republicans if they were randomly selected is (Round to four decimal places as needed) b. The mean of this distribution is (Round to three decimal places as needed.) The standard deviation of this distribution is (Round to three decimal places as needed)

3 Minimum of IID exponentials Let Z1...., Zn be IID exponential random variables with mean 8. That is, each Zi has a PDF given by: $(=) = exp(-/B1, where 2 and 8 are positive. Derive the probability density function for min(Z.....2) (ie, the minimum of random variables 21,..., Zn). You should find that the probability density function for min(21..., Zn) is that of an expo nential random variable. What is the mean of min(2...., 2..)?

Suppose that the average time a fully charged 6 volt laptop battery will operate a computer is 20 hours and follows the exponential probability distribution. Determine the following probabilities a) Determine the probably that the next charge will last less than 1.5 hours b) Determine the probability that the next charge will last between 2.7 and 48 hours c) Determine the probability that the next charge will last more than 3 hours a) The probability that the next charge will last less than 1.5 hours is (Round to four decimal places as needed.) b) The probability that the next charge will last between 27 and 48 hours is (Round to four decimal places as needed.) c) The probability that the next charge will last more than 3 hours is Round to four decimal places as needed.) A random variable follows the continuous uniform distribution between 35 and 145. a) Calculate the probability below for the distribution. P(55 SX S115) b) What are the mean and standard deviation of this distribution? a) P(55 sxs 115) = (Type an integer or decimal rounded to three decimal places as needed.) b) The mean of this distribution is (Type an integer or decimal rounded to two decimal places as needed.) The standard deviation of this distribution is (Type an integer or decimal rounded to two decimal places as needed.)

A component's lifetime has an exponential distribution with an expectation of T = 4100 hours. What is the probability it will still be running when the time is 4600 hours? Sate your answer rounded to three decimal places.

Problem 7. (0%) (Sec. 8.4 Transformations of Two Random Variables) and X, be two independent exponential random variables with the identical mean 1/1 Let Y1 = X1 and Y2 = X1 + X2. Use Theorem 8.8 (on pp. 356 - 357 of the textbook) to find the probability density function of Y, by the following steps. (a) (0%) Find fxx, (21,22), the joint probability density function of X and X2. (b) (0%) Find fyı,ya (41, 42), the joint probability density function of Yį and Y2. (c) (0%) Find fy.(y), the probability density function of Y2.

A component's lifetime has an exponential distribution with an expectation of T = 3100 hours. What is the probability it will fail prior to the time of 2900 hours? Sate your answer rounded to three decimal places.

A component's lifetime has an exponential distribution with an expectation of T = 1900 hours. What is the probability it will fail prior to the time of 2600 hours? Sate your answer rounded to three decimal places.