1. The input to a communication channel is a random variables x and the output is another random variable y. The joint probability mass functions of x and y are listed in the following. Find P{y=1|x=1}, P{x=1|y=1}, and pxy. y=-1 y=0 y=1 x= -1 1/4 0 0 X=0 1/8 1/4 1/8 x = 1 0 0 1/4
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- Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…Suppose X and Y are the random variables with joint PMF given by: X/Y 11.2 5.75 -3 0 0.07 -2 0.2 0.08 -1 0.1 0.1 0 0.15 0.1 1 0 0.2 a.) Compute E(X2)Let X be a random variable with probability mass function P ( X = 1 ) = 1/2 , P ( X = 2 ) = 1/3 , a n d P ( X = 5 ) = 1/6 . Then E[1/x]=?
- 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-Y2Let i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tablesLet X1 and X2 be two independent random variables. Suppose each Xi is exponentially distributed with parameter λi. Let Y=Min (X1, X2). A) Find the pdf of Y. B) Find E(Y). Hint: Let Y = Min (X1, X2). 1. P[Y > c] = P[Min (X1, X2) > c] = P[X1 > c, X2 > c] 2. Obtain the pdf of Y by differentiating its cdf of Y.