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EBK AN INTRODUCTION TO MATHEMATICAL STA
- Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X and Y have the joint densityarrow_forwardConsider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)arrow_forwardConsider 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…arrow_forward
- (b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.arrow_forwardFor a random variable (X) having pdf given by: f(x) = (k)x^3 where 0 ≤ x ≤ 1, compute the following: a) k b) E(X). c) Var(X). d) P(X > 0.25).arrow_forwardLet X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?arrow_forward
- Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2arrow_forwardIf the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.arrow_forwardThe random variables X and Y have the joint density: fX,Y(x,y) = 2−x−y, for 0<x<1, 0<y<1 0, otherwise For each of the following, please provide your answers in three decimal places: (a) What is the expected value of X? (b) What is the variance of X? (c) What is the covariance of X and Y? (d) What is the correlation of X and Y?arrow_forward
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