X (f) = Ag cos ogi+Bo - sin wn t, where og is a real constant. Find the power %3D density spectrum of X (t), if A, and Bo are uncorrelated random variables with zero mean and same variance.
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- Compute Var(ys) for the following model, where e; ~ wn(0, 0.01), i.e., a white noise process with mean zero and variance 0.01. 4 = 1+0.5y-1 + et, yo = 1. Please give the exact answer.2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.Consider an estimator of μ: W= 1/6 Y1+1/16Y2+1/4Y3+1/8Y4+1/2Y5 This is an example of a weighted average of the Yi's. Show that W is also an unbiased estimator of μ. Find the variance of W.
- Suppose that X, Y are independent standard normal RVs. Find the joint pdf of Z, W where Z = XY, W = 3X – 2Y.X1,X2,...,Xn are i.i.d. random variables from N(0,θ) distribution. Here, θ= σ2 >0is the variance of the distribution, and is an unknown parameter. Find the MLE of θ(no need tocalculate the second derivative).Let X and Y be independent Gaussian random variables, each distributed according to (0, σ2). 1. Find the joint density function of the random variables Z = X + Y and W = 2X −Y . What is the correlation coefficient between these two random variables
- Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.Let Ap and B be the random variables. A random process is defined a X () = Ag cos ont+ B sin ant, where og is a real constant. Find the power density spectru m of X (), if Ag and B, are uncorrelated random variables with zero mean and same variance.Suppose X and Y are independent, exponentially distributed random variables with rate parameter λ, λ > 0. Find the joint PDF of U and V , where U = X + Y, V = X/Y.
- 1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.Let X be a Gaussian RV on with expected value (mean) E[X] = 1, Var[X] = 1. Let Y = -3(X-1). Compute the expected value of Y. QUESTION 7 Let X be a Gaussian RV on with expected value (mean) E[X] = 1, Var[X] = 1. Let Y = -3(X-1). Compute the variance of Y.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?