Total marks 16 5. Let (,,P) be a probability space and let X : → R be a random variable whose probability density function is given by f(x) = }}|x|e¯|×| for x Є R. (i) (ii) Find the characteristic function of the random variable X. [8 Marks] Using the result of (i), calculate the first two moments of the random variable X, i.e., E(X") for n = 1, 2. (iii) What is the variance of X? [6 Marks] [2 Marks]
Q: I keep solving this and the solutions that I have been attempting do not seem to be correct, what…
A: Part a) Notation for the PMF and Probability of Getting 7 Non-Skin Gifts Before the Second Desired…
Q: Prove: ∃sets A, B ∀functions f : A →B, f is a bijection.
A:
Q: Use the given information to fill in the number of elements for each region in the Venn diagram.…
A:
Q: (a) What is the probability that a five-card poker hand contains exactly one Ace? (b) Find the…
A: Probability is a measure of how likely an event will occur. To calculate probability, the formula is…
Q: Please solve the attached
A: To find the required sample size:1. Look up the Z-score for 99% confidence in the provided table.…
Q: Final Answer rounded to 4 decimal places
A: To solve this problem, let's first break down the steps:Count the Total Candies: We need to…
Q: Given a normal random variable X with mean µ = 20 and standard deviation σ = 3 and a random sample…
A:
Q: Please solve the attached
A: In Bayesian statistics, understanding uncertainty and updating beliefs based on observed data is…
Q: Assignment 2 1. A bag contains 10 red balls and 15 blue balls. If 6 balls are drawn without…
A: Detailed explanation:Problem 1: Probability of Drawing 3 Red BallsGiven:N=25 (total balls)K=10…
Q: Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a…
A:
Q: 4-2. 1 A calculator with a random number generator produces the following sequence of random…
A: a)To find the sample mean, add all values and then divide it by the number of values.…
Q: 6. This question is a simple exercise in changing variables. A gun fires at random in the angular…
A: Part 1: Show that g(y)dy=π11+(y/l)21ldyWe start with the setup. A gun fires randomly in the…
Q: Consider a large-sample level 0.01 test for testing Ho: p = 0.2 against Ha: p > 0.2. USE SALT (a)…
A:
Q: Prove that if events A and B are independent, then...(a) A and Bc are independent. (Hint: Split up…
A:
Q: B1. Let X be a continuous random variable with PDF 2 f(x) = ½ ½ (2 - x) 9 and f(x) = 0 otherwise.…
A: (a) Find the value of the constant cFor f(x) to be a valid probability density function (PDF), the…
Q: Which of the following are true about the normal curve? Check all that apply. The normal curve…
A: The normal curve, also known as the Gaussian curve or bell curve, is a type of continuous…
Q: 40 25 Fig 2 75 20 50 Fig 3 75 40 50 28 45 Figures 1 to 3 each shows a top view and a front view of…
A: Step 1:
Q: 1). The p-value for this test is . 072. Do we reject the null hypothesis?2). Is there sufficient…
A: The null hypothesis is a general statement or default position that there is no relationship between…
Q: Zane owns a restaurant, which features wings and ice cream. He is considering two new flavors of…
A: Hope this help. For part b which is not given, just use the venn diagram above.
Q: Driver Killed Driver Not Killed Seatbelt Used 36 70 Seatbelt Not Used 44 30 (a): If one of…
A: Let's analyze the given data in a table: Driver KilledDriver Not KilledTotalSeatbelt…
Q: Need help with probability
A:
Q: Readers of the magazine Popular Science (August 1990) were asked to phone in (on a 900 number) their…
A: The poll conducted by Popular Science in 1990, where readers were asked to phone in their responses…
Q: Suppose X and Y are r.v.s such that E(X) = 4, Var(X) = 48, E(Y ) = 2, andVar(Y ) = 5. Find upper and…
A: To find the upper and lower bounds for E(XY), we need to use the Cauchy-Schwarz inequality. For any…
Q: Let X and Y be i.i.d. Expo(λ), and let T = ln(X/Y ).(a) Use change of variables to find the joint…
A: That is the standard form of the logistics contribution.
Q: If X is a normal random variable with mean μ = 3 and standard deviation σ = a. P(X > 11) b. P(2-3|X…
A: Given mean(μ) = 3 and standard deviation (σ) = 4 a) P(x > 11) Since Z=σx−μ Z=411−3 Z = 8/4 = 2…
Q: g12 math data management 10
A: Step 1:Here are given 8 countries in race.The top 3 countries get gold, silver, and bronze.Need…
Q: 1-6.5 Two solid-state diodes are connected in series. Each diode has a probability of 0.05 that it…
A: Step 1:write the sample space Step 2:calculate number of elements of each event Step 3:find the…
Q: 13. A committee of 12 is to be randomly selected from 15 science teachers and x math teachers. If…
A: Step 1:number of teachers to be selected =12number of available science teacher =15number of…
Q: In a Poisson process with A = 500 arrivals per day, find the approximate probability that the…
A: In a Poisson process, the number of arrivals per unit time follows a Poisson distribution, and the…
Q: Answer the question in the attached photo.
A: Step 1: Step 2: Step 3: Step 4:
Q: Suppose that rainfall Y in central Indiana has the log-normal distribution,2(2.25, =0.05) YLNμσ=a)…
A: To solve the given problems for a log-normal distribution with parameters mu = 2.25 and sigma^2 =…
Q: Answer rounded to 4 decimal pla
A: To find the probability that the 3 dogs are all side by side when arranging 14 animals (including 3…
Q: The manager of a pizza parlor open 7 Days a week allows each of three employees to select one day…
A: Step 1:The manager of a pizza parlor open 7 Days a week allows each of three employees to select one…
Q: suppose the probablity of heads on a biased coin 1/3 the coin is flipped 7 times. define x as the…
A:
Q: Help me solve # 4
A: Probability is a measure of how likely an event will occur. For a probability distribution, the sum…
Q: 3. CLO4 Let X be a continuous random variable with probability density function (pdf) Ix(x)= (2r…
A: (a): Obtain the marginal probability density function of ( Y ).Given ( X ) has a probability density…
Q: A supplier of paper produces 10-sheet packs of certain high gloss paper for the production of…
A: To find the mean and standard deviation of Xi, where Xi is the number of sheets of paper in the i-th…
Q: 41. 1. If X is a normal N(4, 64) distribution, find (a) P(6≤ X ≤14) (b) P(X ≤ -5.2) 2. Given a…
A: Step 1:if X is random variable follow normal distribution it represented as X ~ N (μ,σ2)…
Q: 43. μ 1. (Confidence interval for µ given 02²) Determine the confidence interval [01, 02] μ,0.95…
A: Question 1)We are given the following:sample mean , x̄ = 3sample size, n = 400confidence level, c =…
Q: g12 math data management 3
A:
Q: Probability Trees are closely related to decision trees, which are used in finance and other fields…
A: Let C : be the event that the product is manufactured by CilicoA : be the event that the product is…
Q: DURPA batteries have an average lifetime of 3years and a variance of 0.25 years2. If we assume that…
A: Given Data:mean (µ)=3 yearsvariance (σ2)=0.25 years2 The standard deviation is the square root of…
Q: 3. (i) State precisely what it means that the sequence (x (n)) DEN Converges to the limit xЄ R². [5…
A: In the context of sequences, convergence refers to the property that a sequence has if its terms get…
Q: 1. A multiple-choice test has 10 questions, each with 4 choices. If a student guesses randomly on…
A: Step 1:(1)A multiple-choice test has 10 questions.n=10Each question has 4 choices.Probability of a…
Q: Handwriting
A: Step 1:the given data set:33 , 28 , 27 ,30, 63 , 89 ,52 , 36 27 , 9, 92 ,56 ,42 , 67 , 88 The mode…
Q: The probabilities and gain or loss for guessing on a standardized test is shown in the table to the…
A: The given values are:Probability of guessing correctly (P(C)) = 1/5Probability of guessing…
Q: Final answer rounded to 2 decimal places
A: Here, we are given a frequency table showing the number of sick days and their frequencies, and we…
Q: Traditionally, the game of Rock, Paper, Scissors is played with two people. Each of the two people,…
A: To find the joint probability mass function (PMF) of R, P, and S—where R represents the number of…
Q: Traditionally, the game of Rock, Paper, Scissors is played with two people. Each of the two people,…
A: **Step 1: Understand the Outcomes**Each player can choose among three options (rock, paper,…
Q: Please break it down so that as I compare my work, I am able to identify my mistakes.
A:
![Total marks 16
5.
Let (,,P) be a probability space and let X : → R be a random
variable whose probability density function is given by f(x) = }}|x|e¯|×| for
x Є R.
(i)
(ii)
Find the characteristic function of the random variable X.
[8 Marks]
Using the result of (i), calculate the first two moments of the
random variable X, i.e., E(X") for n = 1, 2.
(iii) What is the variance of X?
[6 Marks]
[2 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2Ff9a374bb-d36d-408f-b888-4c0c9c073c6a%2F3m8z5sb_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 7 images
- The continuous random variable XX has a probability density function (pdf) given by Part(d) Find Var(X), correct to 2 decimal places Part(e) Find the median of X, correct to 2 decimal places.Part(f) Find E(X), correct to 2 decimal placesThe continuous random variable X has a probability density function (pdf) given by [1- x for 0 ). Give your answer as a decimal, correct to 2 decimal places. Part(f) Find the value of c correct to 2 decimal places given that E(X+c) = 4E(cX).10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.
- Tossa coin once, S = (H,T) with P,(H) = p and P,(T) = q. Define X(-) by X(H) = ₁ X(T) = 0. Compute and plot the probability distribution function F, (x).Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 otherwise. Q2(i.) Sketch(Shade) the support of (X1, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint…The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.
- The continuous random variable XX has a probability density function (pdf) given by Part(a) Find the value of qq such that P(X<q)=1/4. Give your answer as a decimal correct to 3 decimal places. Part(b) Two independent observations of X are taken. Find the probability correct to 2 decimal places that one is less than 1/2 and the other is greater than 1/2. Part(c) Find P(X>12). Give your answer as a decimal, correct to 2 decimal placesSketch the graph of the probability density function over the indicated interval. f(x) = 2x/25, [0, 5] Find the indicated probabilities. a) P(0 < x < 3) b) P(1 < x < 3) c) P(3 < x < 5) d) P(x ≥ 1)Sketch the graph of the probability density function over the indicated interval. F(x) = 1/20, [0, 20] Find the indicated probabilities. a) P(0 < x < 12) b) P(8 < x < 12) c) P(13 < x < 20) d) P(x ≥ 5)
- 4. (i) Let a be a positive constant and f(x) = ax² e −4x x = R. Find a such that f(x) is a probability density function. [6 Marks] (ii) Let X be a random variable with probability density function in (i) (a) Find (A), the characteristic function of the random variable X. (b) Using (A), calculate E(X) and Var(X). [15 Marks] [14 Marks]Let X be a continuous random variable whose probability density function is: f(x)- 1.5x2 for -1A continuous random variable X has probability density function f given by (kx* (4-x) for 0sx54 f(x)={ elsewhere Find (a) the value of k. (b) the mean and variance of X. (c) P(lSEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON