i.i.d means independent and identical distributed
i.i.d means independent and identical distributed 7. Let Xi , X, , ,X, beii.d. r.ν. with...
Problem 7. Let Xi, X2,..., Xn be i.i.d. (independent and identically distributed) random variables with unknown mean μ and variance σ2. In order to estimate μ and σ from the data we consider the follwing estimates n 1 Show that both these estimates are unbiased. That is, show that E(A)--μ and
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W) (5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 11. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt < X <Y). (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? (e) Compute PIZ> t where t20. (f) Compute the pd.f. of Z. Z = min(X,Y)
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u. (7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
Exercise 7. Let Xi, X2, . . . be independent, identically distributed rundorn variables uithEX and Var(X) 9, and let Yǐ = Xi/2. We also define Tn and An to be the sum and the sample mean, respectively, of the random variablesy, ,Y,- 1) Evaluate the mean and variance of Yn, T,, and A (2) Does Yn converge in probability? If so, to what value? 3) Does Tn converge in probability? If so, to what value? (4) Does An converge...
Let X1, ..., Xn be i.i.d. [Recall that i.i.d. stands for independent and identically distributed.] Since X1, ..., Xn all have the same distribution, they have the same expected value and variance. Let E(X1) = µ and V ar(X1) = σ 2 . Find the following in terms of µ and σ 2 . (a) E(X2 1 ). Note this is not µ 2 ! (b) E( Pn i=1 X2 i /n). (c) Now, define W by W = 1...
i.i.d is independent, identical distributed. Please solve this problem and write clearly. I will give a good review for sure. Thanks!
Problem 3 Let Xi, X2,... , Xn be a sequence of binary, i.i.d. random variables. Assume P (Xi 1) P (Xi = 0) = 1/2. Let Z be a parity check on seluence Xi, X2, ,X,, that is, Z = X BX2 e (a) Is Z statistically independent of Xi? (Assume n> 1) (b) Are X, X2, ..., Xn 1, Z statistically independent? (c) Are X, X2,.., Xn, Z statistically independent? (d) Is Z statistically independent of Xi if P...