4. Let the two dimensional discrete random variable (X,Y) have the following joint probability distribution. Y 0 1 2 X 1 2p 0.08 0.05 2 0.1 0.1 р 3 0.25 0.08 0.13 d. Find the marginal distribution of X for a given Y=2 ief(XIX = 2)
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- Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)If Y is a continuous, uniformly distributed random variable over the interval(4,10), then the value of the PDF between 4 and 10 is?Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure of the right and left tires are random variables X and Y respectively, with the joint p.d.f. f(x, y) = K(x^2+ y^2) when 20 < X < 30 and 20 < y < 30 and zero otherwise. Determine the marginal distribution of air pressure in the right tire alone. (Leave it in terms of K)
- 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, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?Suppose X is a continuous random variable with a triangular distribution defined over 0 to 2. Picture two right triangles pushed together with a peak at 1. f(x) = 0 for x < 0 f(x) = x for 0 < x < 1f(x) = 2 - x for 1 < x < 2f(x) = 0 for x > 2a. Calculate E(X).b. Calculate Var(X)c. Let G(X)=8+3X. What is the E[G(X)] and the variance of G(X)?
- Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?