Total marks 16 5. Let (N,F,P) be a probability space and let X : N → R be a random variable such that the probability density function is given by f(x)=ex for x € R. (i) Find the characteristic function of the random variable X. [8 Marks] (ii) 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]
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![Total marks 16
5.
Let (N,F,P) be a probability space and let X : N → R be a
random variable such that the probability density function is given by
f(x)=ex for x € R.
(i)
Find the characteristic function of the random variable X.
[8 Marks]
(ii) 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%2F48cace1e-149b-4050-a3e0-07541c51450a%2Fp6uixts_processed.jpeg&w=3840&q=75)
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- The 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).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 places
- 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.Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?Let X be a random variable with probability density function 3 f(x) = ,a², –1 < = <1 2 Which of the following is the probability density function of Y = X – 1?
- 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 following function represents joint probability density function for the random variables X and Y. Find the covariance of X and Y. f(x,y) = }0 S6xy²,0 < x < 2 0Let X be a random variable with probability density function 0 < a < 1 f(2) = { 2 otherwise Calculate P(10-4Daily total solar radiation for a specified location in Florida in October has probability densityfunction given byf (y) =$ (3/32)(y − 2)(6 − y), 2 ≤ y ≤ 6,0, elsewhere,with measurements in hundreds of calories. Find the expected daily solar radiation for October.The probability density function of a mixed random variable X is as follows: af(x) 0.20 8(x) 0.108(+) 0,085(x-1) XX a) Find the value of the constant a b) Calculate P[X s 2] 0.30 8(x-3) 3 0.15 8(x-5) 5 XLet X be a random variable with probability density function, What is Var(X)? f(x)=L cx (5-x) for 0 ≤ x ≤ 5 0 otherwise 250/2 125/2 125 Does not existSEE 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