A sinusoidal random process is defined as X[n] = A cos(0.3n) for A~N(0, 1). Compute the mean sequence #x[n]. riance sequence ơn.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Show that the random process X(t) =cos(2π fot + θ) Where θ is an random variable uniformly distributed in the range {0, π/2, π, π/3} is a wide sense stationary process .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
- Suppose that the random variable X is continuous and takes its values uniformly over the interval from 0 to 2. What is P{X = 1.5 or X = 0.4}?Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
- they take samples of 4 fireworks for quality co from and examine them for defects. let X be the number of defective fireworks in the sample of 4.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Consider a random process which is given by Y(t) = t - Z where Z is a random variable with mean 1.2 and second moment 2.5. The autocovariance of the random process X(t) is
- Assume that the variables Y1, Y2,... in a compound Poisson process have Bernoulli distribution with parameter p . Show that the process reduces to the Poisson process of parameter λp.Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Consider a random variable X with E[X] = 10, and X being positive. Estimate E[ln√X] using Jensen’s inequality.