Let x1, x2X. be a random sample of size (n) taken from Poisson( 8), and y = 2x, then F(x1, x2 , X, :0)
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- Let x1, x2 X, he a random sample of size (n) taken from Poisson( 0), and y = E",x, then F(x, x2, Xn :0) giy: 0) F(x1, X2,Let Y₁, Y2, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0,0). Consider = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that is a biased estimator for 0.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
- Suppose the random variable X~ Beta(a, 3), namely its PDF fx(x) = I(a + 3) r(a)(3) -ra-1(1−z)3-1, 0 < x < 1, Beta(a +3, y), namely its PDF fy (y) = I(a +8+y) r(a + 3)(y) ¹(1 − y)*-¹, 0 < y < 1, and X and Y are independent. Define U = XY, V = X. Find the joint PDF fuv(u, v).Let X = (X1, X2)' be a random sample from Po(A), A > 0. Show using the definition that S(X) = X1+ X2 is sufficient for A.Let X1, X2, X3 be independent Exp()-distributed random vairables. Determine the pdf of X(1)X(3)
- Suppose that Y₁ and Y₂ are uniformly distributed over the triangle shaded in the accompanying figure. 3₂ (0, 1) (-1,0) (a) Find Cov(Y₁ Y₂). Cov(Y₁, Y₂) = (b) Are Y₁ and Y₂ independent? Yes O No (1, 0) (c) Find the coefficient of correlation for Y₁ and Y₂. P= y/₁ (d) Does your answer to part (b) lead you to doubt your answer to part (a)? Why or why not? O Even though Cov(Y₁Y₂) # 0, Y₁ and Y₂ are not necessarily dependent. Since Cov(Y₁ Y₂) # 0, we should expect Y₁ and Y₂ to be dependent. O Since Cov(Y₁, Y₂) = 0, we should expect Y₁ and Y₂ to be independent. O Even though Cov(Y₁Y₂) = 0, Y₁ and Y₂ are not necessarily independent.Let f(x, y) = x + y for 0 < x < 1 and 0 < y < 1 The Conditional Variance of Y when X = ; isQ2) A continuous random variable has PDF Kx²+2x+1, -25x5 3. Find K, P(x)>0. X, X² and ¹.
- If(x,,y; ).(x,, y; ),..(x,,y«) be a random sample taken from a logistic regression, then the log-likelihood cquation is, a) ,In| Eln(1-z,) -T, b) Σh(1 -π) Σy n -Σh(π) 1-7, c) i-1 1-1 d) Σyn(1- r)+Σ -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)b) 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.