Let (X; Y) be a continuous random vector with joint probability density function 0.5 -1sx<1,0sysl fx.ylx,v)= else Then P(X > 0) equals
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- Let X and Y be two jointly continuous random variables. Let Z = X² + Y². Show that F2(=) = L Fxr(V=- yř 19) – Fxy(-v/= = y"l»)] fv (w) dy .Suppose that X = (X1, X2, X3) ~ N3(µ, E), where u = (10, 5, 8) and 2, 1 0' Σ= 1 1 0 4 3, What is the distribution of the random vector (X1 + 2X2, X1 – X2)?Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.
- Let x₁ = a +bt² +c+z₁, t= 0, 1, 3,... where a,b and c constants and z, ~ WN a) Let U₁ = x, x4. Find the mean and variance of U,. 1-4 -WN (0,0²)- b) Show that {U}is not stationary. c) Let V₁ = U₁ -U₁. Show that {v} is stationary. d) Find the autocorrelation function for {V,}.Consider the function f(x) = x3 + 5 x² + 3. Find the equation of the normal line at the point (1, 9). 118 a) Oy = 13 13 116 b) Oy = 13 13 c) Oy = 13 x – 4 116 d) Oy = 13 13 O y = -13 x + 22b) Let Z₁ = X-XN (0,1), and W₁ dx YHY~N(0,1), for i = 1,2,3,...,10, then: dy i) State, with parameter(s), the probability distribution of the statistic, T = - 54 1² ii) Find the mean and variance of the statistic T = Σ},wp? Σ1,2,3 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ẞ such that P(T> B) = 0.01, where T = Σ₁Z₁² + ₁ W₁².
- A projectile is launched at an angle theta with respect to the surface with velocity v0 (deterministic). If the angle of inclination is a uniform random variable in [0, pi/2 ], calculate the distribution function of the variable R defined as the point of impact of the projectile on the ground, measured from the origin. Also calculate your expected valueLet X ~ Uniform(-1,1) and Y = X² , find the CDF and PDF.2 Let B be the (k + 1) × 1 vector of OLS estimates. (i) Show that for any (k + 1) × 1 vector b, we can write the sum of squared residuals as = û'û + (ß – b)'X'X(ß – b). Xb)'(y – Xb) = [û + X(ß – b)]'[û + X(ß – b)] and use the fact that {Hint: Write (y X'û = 0.} -
- If Y - Exp(2) and Z ~ Exp(3) are independent, what is pdf of W = Y + Z?Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of Y