Jx (x) = P(1- P) X=1,2,3,.... F, (y,) = ? Fy, (y,) = ? P(Y, <1)=? а. b. С.
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- Let X1,X2,.,X, be a random sample from f(x;8) = 0e-8xIq0,c0) (x). Find a 100y or (1 - a)100 percent confidence interval for the mean of the population.Explain what happens we minimize the following error ftctions to fit a set of data points, E, = E(v, -(4, + 4,x,)) E, = 20, - (4 + 4,x, })* = abs > (y, - (a, + a,x,)) %3! E, E, = E0, - (4, +a,x,))' E, = E(v, - (4, + a.x,))' %3DA random variable Y has a uniform distribution over the interval (8,, e,). Derive the variance of Y. Find E(Y)? in terms of (e,, e,). E(Y)? = Find E(Y) in terms of (e,, e,). E(Y) = Find V(Y) in terms of (8,, e,). V(Y) =
- X,X, and X, is a random sample of size 3 from a population 3. with mean value u and variance o2, T, T, T, are the 3. estimators used to estimate mean value u, where T=X,+X,-X,T,= 2X,+3X, - 4X,& T, ax,+x,+x) 1 (ax,+X,+X,) (i) Are T, and T, unbiased estimators ? 1 2 (ii) Find the value of such that T, is unbiased estimator 3 for u. With this value of A is T, a consistent estimator ? (iii) 3 (iv) Which is the best estimator ?9. A scientist makes two measurements X, Y, considered to be i.i.d. random variables. (a) If X, Y ~ N(0, 1). Find the correlation between the larger and smaller of the values, i.e., Corr(max(X, Y), min(X,Y)). (b) If X, Y 2 Unif(0, 1). Find the correlation between the larger and smaller of the. values, i.e., Corr(max(X, Y), min(X,Y)). DSuppose X, and X, have a bivariate normal distribution. Then, the condition distribution of X, given X, = x, follows a normal distribution with variance, o:(1-p)' of(1-p²) 0,(1-p²) a) b) c) d) 0,(1-p)
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