proof that EX3 : PlooR For general n, we have n!f(x1)f(x2)·…· ƒ(In) which holds for 1
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- Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.A continuous random variable X has the probability density function f(x) so that f(x) is 0 for x≤≤≤≤0; f(x)=x for 0<x ≤ 1; f(x)= 2-x for 1<x≤ 2 and f(x) = 0 for x>2. Find the probability that X is at most 1/4.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.
- LetXbea randomvariable havingthe uniformdistribution ontheinterval(θ,θ+1),θ∈R.Showthat Xis not complete.a. Explain the difference between the method of moment generating function and the method of incarnation to find the probability density function of a random variable b. Let X1, X2, X3 ,,,,, each of them has Poisson distribution with parameter lamda 1, lamda2, lamda3,........ i. Get the probability distribution of Y=X1+X2+....Xn ii. Get the mean and variance of Y c. Y1 and Y2 be a joint probability function as shown below in the screenshot imageU=Y1/(Y1+Y2), Using the method of incarnation (Jacobian), find the probability density function of UA random variable follows a distribution of the form f(x)=k(x+2)e^-x over x>orequalto 0 . Determine the probability that two independent samples are drawn from the population and they both have that 1 < x < 2. State your answer in exact form (with a bunch of e’s), showing all work.
- Statement: Let U be a uniform (0,1) random variable. For any continuousdistribution function f, the random variable X defined by X = F-1(U) has distribution function F.Prove the above statement.Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Var(a + X) = Var (X)
- A service station has both self-service and full-service islands. On each island, there is asingle regular unleaded pump with two hoses. Let X denote the number of hoses beingused on the self-service island at a particular time and let Y denote the number of hoses onthe full-service island in use at that time. The joint probability mass function of X and Y isgiven below: X Y0 1 20 0.10 0.04 0.021 0.08 0.20 0.062 0.06 0.14 0.30a. Find the marginal probability mass function of X and Yb. Give the verbal description the event (X≠0 and Y≠0) and compute the probability of this eventA crazy statistics tutor only passes 10% of students in a university subject, by examining all students in foreign languages that they do not know. If 20 students are randomly assigned to this tutor, what is the probability that at most 1 student passes in the tutorial group? A) 0.392 B) 0.270 C) 0.0 because their mark is a continuous rendom veritable D)0.122(Proof of the independence of X and S2 for n = 2) IfX1 and X2 are independent random variables having thestandard normal distribution, show that(a) the joint density of X1 and X is given by f(x1, x) = 1π · e−x−2e−(x1−x)2for −q < x1 < q and −q < x < q;(b) the joint density of U = |X1 − X| and X is given by g(u, x) = 2π · e−(x2+u2) for u > 0 and −q < x < q, since f(x1, x) is symmetricalabout x for fixed x;(c) S2 = 2(X1 − X)2 = 2U2;(d) the joint density of X and S2 is given byh(s2, x) = 1√π e−x2· 1√2π(s2)− 12 e− 12 s2for s2 > 0 and −q < x < q, demonstrating that X and S2are independent.