Suppose that Y, and Y, are uniformly distributed over the triangle shaded in the accompanying figure. y2 (0, 1) (-1, 0) (1, 0) Find fr, s, s
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Let Y1 and Y2 be independent and uniformly distributed over the interval (0,1). Find P(4Y1<Y2)If X1, X2, ... , Xk have the multinomial distribution ofDefinition 8, show that the covariance of Xi and Xj is−nθiθj for i = 1, 2, ... , k, j = 1, 2, ... , k, and i Z j.
- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Let X be a symmetrically distributed RV with mean =0 It is known that P(X^2<1)=a, where a∈(0,1)a∈(0,1) is an unspecified real number. Find P(X<1) in terms of a?LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
- Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.Suppose X1,…,Xn,…are identically distributed with mean E(X1)=μ<∞ and Var(X1)=σ2<∞. In addition, we assume that Cov(Xk,Xk+1)=0 for k=1,2,… but Cov(Xk,Xj)=0 whenever ∣k−j∣≥2. (a) Find the limiting distribution ofXˉn=n−1i=1∑nXiasn→∞. (b) Find the limiting distribution of Zn=∑nnXin+∑n=1nXiX,eias n→∞. (c) LetY1,Y2… be i.i.d random variables with mean 0 and variance 1 . Additionally, letXk=Yk+Yk+1 for ,k≥1. Find the limiting distribution of Xˉn⋅Suppose X has uniform distribution on (−1, 1) and, given X = x, Y is uniformly distributed on (−(1-x^2)^1/2, (1+x^2)^1/2). Find the join denisty of X and Y.
- If X is uniformly distributed over (1, 2), find z such that P ( X > z+μx ) = 1/4Assume that X1, X2, X3 ∼ Exp (λ) are independent and evenly distributed with the distribution Exp (λ = 1). (a) Determine the distribution of Y = X1 + X2 + X3 and state its PDF f (y). (b) Determine the distribution of U = 2YSuppose {X1, X2, …, X7, Y} are independent RVs. Each Xj is Rayleigh with mean and Y = P(2) is Posson. Let (X1, …, X7) iid and find E(X) 2. Var(X) 3. E(X-2Y) 4. Var(X-2y)
