The density function of a random variable X is =e* for r20 otherwtse fx (x) 3D0 Find (a) E(X) (b) E (X) (c) E [(X – 1)²]
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![The density function of a random variable is
fx (x)
=e* for x20
otherwise
(b) E (X)
Find (a) E(X)
(c) E [(X- 1)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb007501-3105-4092-9424-569223a9ca16%2F41730072-55e3-4d9e-92db-cbbd35c64747%2F0snunyl_processed.jpeg&w=3840&q=75)
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- The density function of a random variable X is gx(x) = [5e-5x elsewhere Find (i) E[X] (ii) E [(X – 1)²1 (iii) E [3X – 1]Q. 5 Let X be a continuous uniform random variable on [-a, a]. Find the pdf for Y = g(X), where 0, x b. g(x) = 3x,1. A random variable X has a probability density function given by 2x + 6 fx(x) -3 < x < 2. 25 Let Y = |X|. (a) What is Sy? (b) Is the transformation one-to-one? (c) Derive the cumulative distribution function of Y. (d) Derive the probability density function of Y.
- (vi) Given below is a joint pdf of X₁ and X2₂ which are independent exponential random variables with parameter X. f(x1, x2) Find the joint probability density function (pdf) of Y₁ = X₁ − X₂ and Y₂ = X₁ + X₂ using the transformation method. =e-21-x2 0 < x1 <∞0,0 < x₂ <∞Let X - Poisson(666). That is, the density of X is -666 (666)* k! P(X = k) = k = 0, 1, 2, .... • (i) Derive E(X) and show all your steps. (ii) Derive E(X²) and show all your steps. • (iii) Derive Std(X) and show all your steps.1. A continuous random variable X is defined by (3+x) f(x) - 3sxs-1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean 2. If the probability density function of random variable is given by f(x) = wch 1srs2 sech x a. Find the mean b. Find the total area 3. If the probability density function of random variable is given by f(x) = sinh 2x 1sxs2 a. Find the mean b. Find the total area c. Illustrate the graph (use bell shaped figure) II II
- iid Let X1,· where f(xi, Find the log-likelihood .function ., Xn * f(x;, 0), • 0) = Q³e-o°zi O 93n e-0°x; 3nln (θ) e- ' Σ 3nln(0) – 0 C–1 *; i=1 3nln(0) – 0šn xi O None: The density function of a random variable X is fx (x) = e* for x2 0 otherwise Find (a) E(X) (b) E (X) (c) E [(X – 1)?]b) Suppose that a sequence of mutually independent and identically distributed continuous random variables X₁, X₂, X3X₁ has the probability density function 1 (x-8)² 2 for -00The probability density function of X is given by X 1 3 4 5 6 P(X) K 3K 5K 7K 9K 11K 13K Find (i) P (X 5) (iii) P (3 0.3.Example 28 : IfX is the continuous random variable whose density function is if 0Show that the following are the probability density functions: fi(x) = e-*I(0,c0) (x) f2(x) = 2e¬*I(o,00) f(x) = (0 + 1)f1 (x) – Of2(x) 0 < 0 < 1SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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