The power spectral density of a stationary random process is given by [A Sxx (@) -K<ω< Κ XX otherwise Find the aucorrelation function.
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- A zero-mean stationary Gaussian random process X(t) has power spectral density S_x(f). Determine the probability density function of a random variable obtained by observing the process X(t) at some time t_kA continuous random variable X has a uniform distribution on the interval [−3,3]. Sketch the graph of its density function.X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
- The probability density function of the random variable X is as in the picture with λ> 0. Find the maximum likelihood estimator (λ^) of the parameter λ.Suppose that the continuous random variable X has CDF Fx(X) = {(x-2)/x , x>2 and 0, x=<2} a. Determine, and sketch, the pdf (probability density function) of X. b. Find the mean and variance of X c. Determine the pdf of the random variable Y=X^2Find the moment generating function MU(t) for the standard uniform random variable U (the continuous random variable whose density function is 1 on [0,1] and 0 elsewhere)
- The probability density function of the random variable X is as in the picture with λ> 0. Find the moments estimator (λ^) of the parameter λ.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.Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.
- The joint PDF of the random variables X and Y is constant on the region (x,y):x≥0∩yleq1∩−0.5≤x−y≤0, shown in the image below, and is zero outside. Determine P(Y>0.5|X<0.5)Suppose the random variable Y has an exponential distribution with mean 2. Use the method of transformations to find the density function of U=(1/5)Y+31. A continuous random variable X has the following pdf: 2. Let X be a continuous random variable with density function