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- Which of the following validates a probability density function?* all f(x) must be greater than or equal to 0 all f(x) must be less than or equal to 1 the integral of the probability density function must be equal to 1 from its lower limit to its upper limit the sum of the probabilities of f(x) must be equal to 1 all of the choices7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).b Lri Consider the probability density fx (x) a.e b L, where X is random variable whose allowable values range from x = (a) the CDF (b) the relation between a and b (c) the probability that x lies between 1 and 2. - o to + 00, Find ∞ to + oo,
- 8)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the joint probability density function of the order statistics X(2), X(4), X(6) for n = 7.13.THEORY OF DISTRIBUTIONSa) Let X₁, X₂, X3, X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x)=2 if 0The probability density function (p.d.f.) of a continuous random variable X is defined to be: kx + for 0 < x < 1 f(x) = |0 otherwise, for some constant k. For these problems, please ensure your answers are accurate to within 3 decimals. Part a) Find the value of k that makes the above function a proper p.d.f. Part b) Find E(X).7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]Determine the conditional probability distribution of Y given that X = 2. Where the joint probability density function is given by f(x,y)= 1 - (x + y) for 1 < x < 4 and 0 < y < 3.b) Suppose that X₁ and X₂ have the joint probability density function defined as f(x₁, x₂) = (x1x²,0x₁51, 0 ≤ x₂ ≤1 elsewhere Find: i) the value of w that makes f(x₁, x₂) a probability density function. ii) the joint cumulative distribution function for X₁ and X₂. iii) P (X₂ ≤ | X₂ ≤ ³).SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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