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Probability And Statistical Inference (10th Edition)
- Show that if ax2+bx+c=0 for all x, then a=b=c=0.arrow_forward(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)arrow_forwardEach front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable-X for the right tire and Y for the left tire, with joint pdf given below. f(x, y) = {K(x². √K(x² + y²) 22 ≤ x ≤ 32, 22 ≤ y ≤ 32 0 otherwise (a) Compute the covariance between X and Y. (Round your answer to four decimal places.) Cov(x, y) = (b) Compute the correlation coefficient p for this X and Y. (Round your answer to four decimal places.) P =arrow_forward
- Suppose that X₁, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x) = 1 – 3¯ª for x ≥ 0 and Fx (x) = 0 for x 1).arrow_forwardThe CDF F(x) of a random variable X is given by; F(x) = { 0, ?? ? ≤ 1 ?(? − 1)^4 , ?? 1 < ? ≤ 3 1, ?? ? > 3 } i) the mean and variance of X.arrow_forwardThe probability of success in a certain game is p. The results in successive trials are independent Determine the generating function of Z and use the result to compute E(Z) and Var(Z) Let Z be the number of turns required in order to obtain r successes.arrow_forward
- b) Let X,, X2,...,X, be a random sample from b(1, p). It is known that X is an p(1-p) unbiased estimator of p and that Var(X) = i. Find the Cramer-Rao lower bound for the variance of every unbiased estimator of p given that I(0) p(1-p) ii. What is the efficiency of X as an estimator of p?arrow_forwardLet X be a continuous random variable symmetric about Y. Let Z = 1 if X >Y OR Z = 0 if X <= Y. Find the covariance of |X| and Z.arrow_forwardIf Var(X1) = 2, Var(X2) = 4, Var(X3) = 3, Cov(X1, X2) = 1, Cov(X1, X3) = -2, and X2 and X3 are independent, find the mean and variance of Y = X1 – 2X2+3X3.arrow_forward
- 2. Let f(x) = 3x(x² − 1) (x³ + 2x + 3) (a) Use distribution to expand the function. (b) Use the rules in from the table above to differentiate the function.arrow_forwardSuppose that X1, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x) = 1 – 2 for x >0 and Fx (x) = 0 for x 1).arrow_forwardQ2: (b) Let X₁, X2, X3 be uncorrelated random variables, having the same variance o². Consider the linear transformations Y₁ = X₁ + X₂, X2 + X3. Find the correlations of Y₁, Y; i, Y₂ = X₁ + X3 and Y3 for i + j. =arrow_forward
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