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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Q: elser (i) E[X] (ii) E [(X – 1)21 (iii) E [3X – 1]
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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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- IfX has the probability density f(#) <1 Find F(x). Select one: O a F(2) = 1- O b. F(r) = 1 - Oc F(e) = = c. F() O d. F()=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.Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is ƒ(x) = 20x³(1 − x) Find the expected value and variance for this random variable. E(X) = = Var(X) 0 ≤ x ≤ 1
- (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.: 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.The continuous random variable X has the following probability density function (pdf), for some positive constant c, for 0 < x < c. 3 (1+x)³ f(x)= (a). Prove that c= √3-1. (a) Accept c = √√3-1. Plot f(x) in the range. (b) Find E(X); Var (X) using numerical integration. (c) Use Inversion method, find 20 random numbers from X =Example 28 : IfX is the continuous random variable whose density function is if 0Suppose that Y₁, Y₂,..., Y is a random sample from a probability density function in the (one-parameter) exponential family so that [a(0)b(y)e-[c(Ⓒ)d(y)], a ≤ y ≤ b, f(y10) = 0, elsewhere, where a and b do not depend on 9. Show that Σd(Y) is sufficient for 0.1. A random variable Y has the probability density function (a) Find E[Y]. (b) Find Var[Y]. [3t-4 t > 1 t< 1 fy(t) = - {3²-4Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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