Given that X is uniformly distributed in the range [1, 3] and Y = X², what is the PDF of the random variable Y?a.O b.C.O d.fy(y)=fy(y) ==fy(y)=fy(y) ==14y0.5¹30.54y0.50.5Y4131≤ y ≤91Y4y0.5"1≤ y ≤90.51≤ y ≤9"1≤ y ≤9

A continuous random variable X is said to follow Uniform distribution with parameters a and b, if…

Answered: Given that X is uniformly distributed…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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)
1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).
If the independent random variables X and Y havethe marginal densitiesf(x) =⎧⎪⎪⎨⎪⎪⎩12for 0 < x < 20 elsewhereπ(y) =⎧⎪⎪⎨⎪⎪⎩13for 0 < y < 30 elsewherefind(a) the joint probability density of X and Y;(b) the value of P(X2 + Y2 > 1).
The joint PDF of the random variables X and Y is constant on the shaded region, as shown in the image below, and is zero outside. It can be determined that fX,Y(x,y)=2/3. Determine E[XY]. (The answer is not 3/4)
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The 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?
If 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.
Q4) If X is a continuous random variable having pdf ke~ (2x+3y) x>0y>0 xy) = = e p(x) { 0 otherwise Find a) the constant k b) P(X>1) ¢) X, X2, 02, standard deviation.
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…
Let 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?
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Given that X is uniformly distributed in the range [1, 3] and Y = X2, what is the PDF of the random variable Y? 1 4y0.5 y0.5 - 1 4y0.5 30.5 4 10.5 4y0.5 a. O b. O d. fy(y) fy(y) fy(y) fy(y) = = = " " 1≤ y ≤9 1≤ y ≤9 1≤ y ≤9 > 1≤ y ≤9