Compute E[X + 2Ỹ].
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Compute E[X + 2Ỹ].
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- The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the for components of X independent random variables?Q2) A continuous random variable has PDF Kx²+2x+1, -25x5 3. Find K, P(x)>0. X, X² and ¹.Suppose X is random variable whose p.d.f. is f2)=(2x-x²), 0, OSxs2 Suppose X is random variable whose p.d.f. is f(x)={4 elsewhere Find the mode , if it exists.
- Suppose that Z₁, Z2, ..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y=)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z, i = 1, ..., .., n. (b) Determine My(t) for the special case that Z₁ ~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.Let X ~ N(0, 2) and Y ~ covariances Cov(X,Y) and Cov(Y, X +Y). Exp(A = 3) be two uncorrelated random variables. Find theSuppose that X,Y are both standard normal random variables and Cov(X,Y)= 1 (a) Compute E (X² – Y²)| (b) Is it possible to compute P (X +Y > 3) from the given information? If so compute, otherwise explain.
- Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)Q. # 4 Suppose k is a binomial random variable with parameters n and p and let x=(k-np)//sqrt(npq) where q=1-p. If x is of o(n^(1//6)) and there exist some positive numbers A,B and C then show that |(b(k;n,p)sqrt(npq))/(varphi(x))-1| < (A)/(n)+(B|x|^(3))/(sqrtn)+(C|x|)/(sqrtn), where b(k;n,p) is probability mass function of a binomial random variable and varphi(x) is the probability density function of a normal random variable.Suppose X₁, X2, ..., X₁ are independent random variables and have CDF F(x). Find the CDF of Y = min{X₁, X₂,, Xn} and Z = max{X₁, X₂, ···‚ Xn}.
- Derive the MGF for an exponential random variable X ~ exp(X) (try the Kernel trick!) t-入 ele' –1) A- t3- Show that, if the random variables X and Y are N(0,0,0,o,r), then 1 a) E{f,(y|x)} = еxp|- 20 (2-r)' o/27(2-r²) 1 b) E{fx(x)fp(y)}=- 27o 14-rSuppose that X₁, X2, X3 are mutually independent random variables with the respective moment = (-1) ². generating functions, Mx, (t) = et², Mx₂ (t) = ²t + 3t², and Mx, (t) = (