== 12 Let the process Yд be defined by Y = X + αX-1 where X, is a white noise process (X is a sequence of uncorrelated random variables with zero mean and variance σ2) with autocorrelation R₁(k) so, k=0 5x (S(K) Ο k = 0 and power spectral density S₁₂ (f) =σ², -12-5-½ a) Find the power spectral density, Sy(f), of Y, 11 E[XY] =0 E[X] - FEDO-> E[XY) EDGED]
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- Show that the characteristic function of a Gaussian random variable with zero mean and variance o² is (= x (@) = exp -0² w² 2If a random process, X(t)= Acos wt + B sin wt is given, where A and B are uncorrelated, zero mean random variables having the variance o, find (a) autocorrelationLet X and Y are 2 independent random variables N (0,1) Questions: how to find E(X), E(Y), E(X^2), E(Y^2)? Thank you
- Suppose that X and Y are random variables where the conditional expected value of Y given X is given by E (Y|X) = 3X + 6 If X has the expected value 0 and variance Var(X)=9 determine the minimal possible value of the variance Var(Y) of Y./i) Let X ~Binom(100, 1/4 ). Find a, b 2 R so that Y = aX + b has expected value 0 and variance 1.(ii) Let X1,X2, . . . independent random variables with Bernoulli( 1/3 )‐distribution. Let nSn := Σ Xi. i=1 Find a, b 2 R so that Y = aS20 + b has expected value 0 and variance 1.An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.
- IfX1 andX2 are the means of independent random samples of sizes n1 and n2 from a normal population with the mean μ and the variance σ ^2 , show that the variance of the unbiased estimator ω·Xbar1+(1−ω)·Xbar2is a minimum whenω=n1/(n1+n2)Example: Suppose that X u = (120, 80) and covariance matrix (X1, X2) has a bivariate Normal distribution with mean 6. 3 Σ= 5 What are the mean and variance of a' X, where a = (1, –1)T?Let X1 be a normal random variable with mean 2 and variance 3 and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. (a) What is the distribution of the linear combination Y = 2X1 +3X,? (b) What is the distribution of the linear combination Y = X, – X,?
- "What is the correlation coefficient for this condition? X is the voltage across a resistor, Y is current flowing through a resistor" positive +1 negative -1 "If X is a random variable and Y = ax + b, what is the multiplication factor of variance of X to get variance of Y?" O 1/a a 1/a^2 a^28. Two independent random variables X1=(-1.0.1) and X2-(0.1) can take the following probability values: P(X1=-1)=p, P(X1=0)=p/2, P(X 1=1) 1-3p/2, P(X_2=0)=p, PIX 2=1)=1-D. If Y-X1 X2, determine the average E(Y). E(Y) =? X₁ = (-1,0, 1) y X₂ = (0, 1) P(X₁ = -1) = p P(X₁ = 0) = { P(X₁ = 1) = 1 - T P(X₂ = 0) = p P(X₂= 1) = 1-p Y = X₁ + X₂Suppose that f (x) = e=* for 0 < x Determine the mean and variance of the random variable.