Let (Ω, F, P) be a probability space and let X : Ω → R be a randomvariable whose probability density function is given by f(x) = 12 |x|e−|x| forx ∈ 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 therandom variable X, i.e., E(Xn) for n = 1, 2. [6 Marks]Total marks 16 (iii) What is the variance of X?
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Let (Ω, F, P) be a probability space and let X : Ω → R be a random
variable whose probability density
2 |x|e−|x| 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(Xn) for n = 1, 2. [6 Marks]
Total marks 16 (iii) What is the variance of X?

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- Let X be a continuous random variable with probability density function 1 f (x) : T(1+ x2) Define Y 4. What is the mean of y?Suppose a number is chosen randomly from the interval [0, 4]. Let X be the value chosen. (So X is a uniform random variable over the interval [0, 4].) (a) What is the probability density function of X? (b) Find E(eX + X³).X1 and X2 are independent random variables with X1 ~ Exp(A1) and X2 ~ Exp(A2). Let M = min{X1, X2}. Determine the probability density function of M.
- 3. Let X, Y be independent, standard normal random variables. (a) What is the probability density function of aX + bY? (b) Compute E[X|X – 2Y] and E[X²|X – 2Y].Let X be a random variable with probability density function [ a + bx², 0The continuous random variable X has the following probability density function: Ax, 04. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]Let T be a right triangle with vertices at (0,0), (0,4), and (4 4). We pick a random point in T and let X be its y-component. (a) Show that the cumulative distribution function for X is F(x) = , for 0 < x < 4. (b) Compute the probability density function for X. (c) What is the probability that X is between 1 and 2?Let X and Y be independent random variables with joint probability density function fxy(x, y) = 1/3 (x + y), 0 < x <= 2, and 0 < y<= 1, and 0 otherwise. The marginal pdf fx(x) is given by O a. O b. O c. O d. (2 +2X)/3 (2 + 2X)/3 (X+1/2)/3 (X+1/2)/3 0 < X<= 2 0< X<= 1 0 < X<= 1 0For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. f(x)= 1/93x2 , [4,7] A) find the mean B)find the standard deviation C)Find the probability that the random variable x is within one standard deviation of the mean. (round to three decimal places as needed)4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = gx (x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be - 1.²0⁰ with a similar definition for Y but g replaced by h and x replaced by y. We also define to be the expectation of XY. E[X] = xg(x) dx, E(XY)= (0,00)x(0,00) Tuf(x,y)dady (0,∞) (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y].Use a Venn diagram. Let P(Z) = 0.44, P(Y) = 0.28, and P(Z U Y) = 0.57. Find each probability. (a) P (Z'n Y') (b) P(Z'U Y) (c) P (Z' U Y) (d) P (Zn Y')SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON