Finding marginals with respect to X and Y
1. Let the random vector \((X, Y)\) have the joint density function (continuous case) \(f(x, y)=\left\{\begin{array}{ll}x y e^{-y-x}, & x>0, y>0 \\ 0, & \text { elsewhere }\end{array}\right.\)
Compute the following:
a) \(g_{X}(x)\) (The marginal with respect to \(\mathrm{X}\) )
b) \(h_{T}(y)\) (The marginal with respect to \(Y\) )
Homework Answers
0 The joint p.d.f of X and Y is given by c(1-y), 0<x<y<1 f(x, y) =...
The joint p.d.f of \(X\) and \(Y\) is given by$$ f(x, y)=\left\{\begin{array}{ll} c(1-y), & 0 \leq x \leq y \leq 1 \\ 0 & \text { otherwise. } \end{array}\right. $$Determine the value of \(c\). Find the marginal density of \(X\) and the marginal density of \(Y\) Find the conditional density of \(X\) given \(Y\). Are \(X\) and \(Y\) independent? Why? Find \(E(X-2 Y)\).
The given input signal for 2.7.2 is: x(t) = 3 cos(2 π t) + 6 sin(5 π t). Plz explain ...
The given input signal for 2.7.2 is: x(t) = 3 cos(2
π t) + 6 sin(5 π t).Plz explain steps.Given a causal LTI system described by the differential equation find \(H(s),\) the \(\mathrm{ROC}\) of \(H(s),\) and the impulse response \(h(t)\) of the system. Classify the system as stable/unstable. List the poles of \(H(s) .\) You should the Matlab residue command for this problem.(a) \(y^{\prime \prime \prime}+3 y^{\prime \prime}+2 y^{\prime}=x^{\prime \prime}+6 x^{\prime}+6 x\)2.7.2 The signal \(x(t)\) in the previous problem is...
1. Let X and Y be continuous random variables with joint pr ability density function 6e2re5y İfy < 0 and x < otherwise. y, fx,y (z,y) 0 (a) [3 points] Show that the marginal density function...
1. Let X and Y be continuous random variables with joint pr ability density function 6e2re5y İfy < 0 and x < otherwise. y, fx,y (z,y) 0 (a) [3 points] Show that the marginal density function of Y is given by 3es if y 0, 0 otherwise. fy (y) = (b) |3 poin s apute the marginal density function of X (c) [3 points] Show that E(X)Y = y) =-y-1, for y 0 (d) 13 points] Compute E(X) using the...
Let X and Y be joint continuous random variables with joint density function f(x, y) =...
Let X and Y be joint continuous random variables with
joint density function
f(x, y) = (e−y
y
0 < x < y, 0 < y, ∞
0 otherwise
Compute E[X2
| Y = y].
5. Let X and Y be joint continuous random variables with joint density function e, y 0 otwise Compute E(X2 | Y = y]
Bivariate normal distribution
Problem \(1 \quad\) Bivariate normal distributionAssume that \(\boldsymbol{X}\) is a bivariate normal random variable with$$ \boldsymbol{\mu}=E \boldsymbol{X}=\left(\begin{array}{l} 0 \\ 2 \end{array}\right) \quad \text { and } \quad \Sigma=\operatorname{Cov} \boldsymbol{X}=\left(\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right) $$Let$$ \boldsymbol{Y}=\left(\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right)=\left(\begin{array}{lr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{array}\right) \boldsymbol{X} $$a) Find the mean vector and covariance matrix of \(Y\). What is the distribution of \(Y ?\) Are \(Y_{1}\) and...
3. Let the joint probability density function of W, X, Y, and Z be for,x, y, z) = elsewhere (a) Find the marginal joint probability density function fw.x(w, z). (b) Use part (a) to compute P(O<...
3. Let the joint probability density function of W, X, Y, and Z be for,x, y, z) = elsewhere (a) Find the marginal joint probability density function fw.x(w, z). (b) Use part (a) to compute P(O< W<X<1).
3. Let the joint probability density function of W, X, Y, and Z be for,x, y, z) = elsewhere (a) Find the marginal joint probability density function fw.x(w, z). (b) Use part (a) to compute P(O
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3....
Let the random variable X and Y
have the joint probability density function.
fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
4. Let X and Y be continuous random variables with joint density function f(x, y) =...
4. Let X and Y be continuous random variables with joint density function f(x, y) = { 4x for 0 <x<ys1 otherwise (a) Find the marginal density functions of X and Y, g(x) and h(y), respectively. (b) What are E[X], E[Y], and E[XY]? Find the value of Cov[X, Y]
Let X and Y be two continuous random variables having the joint probability density 24xy, for...
Let X and Y be two continuous random variables having the joint probability density 24xy, for 0 < x < 1,0<p<1.0<x+y<1 0, elsewhere Find the joint probability density of Z X + Y and W-2Y.
A particle moving in one dimension is described by the wave function ...
A particle moving in one dimension is described by the wave function$$ \psi(x)=\left\{\begin{array}{ll} A e^{-\alpha x}, & x \geq 0 \\ B e^{\alpha x}, & x<0 \end{array}\right. $$where \(\alpha=4.00 \mathrm{~m}^{-1}\). (a) Determine the constants \(A\) and \(B\) so that the wave function is continuous and normalized. (b) Calculate the probability of finding the particle in each of the following regions: (i) within \(0.10 \mathrm{~m}\) of the origin, (ii) on the left side of the origin.
The joint p.d.f of \(X\) and \(Y\) is given by$$ f(x, y)=\left\{\begin{array}{ll} c(1-y), & 0 \leq x \leq y \leq 1 \\ 0 & \text { otherwise. } \end{array}\right. $$Determine the value of \(c\). Find the marginal density of \(X\) and the marginal density of \(Y\) Find the conditional density of \(X\) given \(Y\). Are \(X\) and \(Y\) independent? Why? Find \(E(X-2 Y)\).
The given input signal for 2.7.2 is: x(t) = 3 cos(2
π t) + 6 sin(5 π t).Plz explain steps.Given a causal LTI system described by the differential equation find \(H(s),\) the \(\mathrm{ROC}\) of \(H(s),\) and the impulse response \(h(t)\) of the system. Classify the system as stable/unstable. List the poles of \(H(s) .\) You should the Matlab residue command for this problem.(a) \(y^{\prime \prime \prime}+3 y^{\prime \prime}+2 y^{\prime}=x^{\prime \prime}+6 x^{\prime}+6 x\)2.7.2 The signal \(x(t)\) in the previous problem is...
1. Let X and Y be continuous random variables with joint pr ability density function 6e2re5y İfy < 0 and x < otherwise. y, fx,y (z,y) 0 (a) [3 points] Show that the marginal density function of Y is given by 3es if y 0, 0 otherwise. fy (y) = (b) |3 poin s apute the marginal density function of X (c) [3 points] Show that E(X)Y = y) =-y-1, for y 0 (d) 13 points] Compute E(X) using the...
Let X and Y be joint continuous random variables with
joint density function
f(x, y) = (e−y
y
0 < x < y, 0 < y, ∞
0 otherwise
Compute E[X2
| Y = y].
5. Let X and Y be joint continuous random variables with joint density function e, y 0 otwise Compute E(X2 | Y = y]
3. Let the joint probability density function of W, X, Y, and Z be for,x, y, z) = elsewhere (a) Find the marginal joint probability density function fw.x(w, z). (b) Use part (a) to compute P(O< W<X<1).
3. Let the joint probability density function of W, X, Y, and Z be for,x, y, z) = elsewhere (a) Find the marginal joint probability density function fw.x(w, z). (b) Use part (a) to compute P(O
Let the random variable X and Y
have the joint probability density function.
fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
4. Let X and Y be continuous random variables with joint density function f(x, y) = { 4x for 0 <x<ys1 otherwise (a) Find the marginal density functions of X and Y, g(x) and h(y), respectively. (b) What are E[X], E[Y], and E[XY]? Find the value of Cov[X, Y]
Let X and Y be two continuous random variables having the joint probability density 24xy, for 0 < x < 1,0<p<1.0<x+y<1 0, elsewhere Find the joint probability density of Z X + Y and W-2Y.
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