Find: fx,x(x, y) = { cay² 0≤x≤ 1,0 ≤8≤1 otherwise a) Constant c. b) P[X > Y], P[Y < X²]. c) P[min(X,Y) ≤ 1/2], P[max(X,Y) < 3/4]. d) Find the marginal PDFs fx(x) and fy(y). e) Are these random variables independent?
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Let X and Y be continuous random variables with joint
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- a) Find the marginal pmfs of X and Y b) Find the conditional pmf of X given Y = 1A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).Find the linearization of f(x)=1−∫3x+124/(3+t)dt at x=2. upper limit of integral = x+1 lower limit of integral = 3
- Let X and Y have the joint PDF f(x,y) = 2e−x−y for 0<x<y<∞. Find the joint PDF of Z = 2X and W = Y − X, as well as the two marginal PDFs.Approximate the critical points of g(x) = x cos−1 x and estimate the maximum value of g.a. What is the joint marginal PDF of X and Z?b. What is [P 0 <= Y <= 3/2 | X = 1/3 Z = 5/2]?
- If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?A statistician has to decide on the basis of a singleobservation whether the parameter θ of the density f(x) =⎧⎨⎩2xθ 2 for 0 < x < θ0 elsewhere equals θ1 or θ2, where θ1 < θ2. If he decides on θ1 when theobserved value is less than the constant k, on θ2 when theobserved value is greater than or equal to the constant k,and he is fined C dollars for making the wrong decision,which value of k will minimize the maximum risk?Find critical point (0,0) of f(x,y) =e^[(-2x^2)+(3y^2)] is a min, max, or saddle point using the second partials test.
- Suppose that X and Y are independent measurement of a quantity μ. E(X) = E(Y) = μ, σX ≠ σY, Z = αX + (1 - α)Y, α ∈ [0, 1] (a) Show that E(X) = μ. (b) Find α in terms of σX and σY to minimize V(Z). (c) Under what circumstance is it better to use the average (X+Y)/2 than either X or Y alone?Don't the original bounds of f(x,y) come into play when solving for the E(XY) and the marginal pdf of either x or y? image below ...