The power density spectrum of a WSS process is Sw (ω) -4π δ(ω) + 3π δ(ω - 5π) + 3πδ(ω+5π) +2π δ(ω- 4)+2πδ(ω+ 4) (а) What are the frequencies in X(t)? (b) Find mean variance and average power of X(t).
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- A discrete random variable X has probability mass function X 0 1 2 3 4 P(x) 0.1 0.2 0.2 0.2 0.3 Use the inverse transform method to generate a random sample of size 1000 from the distribution of X. Construct a relative frequency table and compare with the theoretical values.1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]continuous random variables Z is known to have an Erlang (2,1.3) type PDF. what is the variance of Z ?
- Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxyX1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is belowLet x be a random n-vector with mean mu and positive definite variancecovariance matrix Sigma. Define Sigma^(-(1)/(2))=(Sigma^((1)/(2)))^(-1). (a) Show that Sigma^(-(1)/(2))(x-mu) has mean 0 and variance-covariance matrix I. (b) Determine E[(x-mu)^(T)Sigma^(-1)(x-mu)] and var[(x-mu)^(T)Sigma^(-1)(x-mu)], assuming for the latter that the skewness matrix Lambda and excess kurtosis matrix Omega of x exist.
- 13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and CovarianceLet random sample from a Poisson distribution with parameter . A random variable X with Poisson distribution has a probability mass function (pmf) of p(x; lambda)=((lambda^x)(e^-lambda))/x!, x=0,1,... with E(X)=lambda and Var(X)=lambda . (d) Find the Fisher information for the random sample of the parameter lambda. (e) What is the Cramer Rao Lower bound for an unbiased estimator of the parameter lambda? (f) Is the MLE of part (c) a minimum variance unbiased estimator (MVUE) of lambda?Let x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.
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