THEOREM 3.3. If X EXP (^,), Y` EXP 2), X and Yare in- dependent, and A + A2, then Z = X + Y has a two-stage hy- poexponential distribution with parameters , and A2; that is, Z HYPO (A1, A2).
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- Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to thisLet X be a point randomly chosen on a stick of length 1 with a uniform distribution. Let Y be a point chosen between 0 and X on this stick with a uniform distribution. Y X 1 Find the covariance between X and Y. Please provide the solution step by step.Theorem 3: Let x - (x1,X2 -- . Xp . .., x,) be a data set with a sample mean ĩ and a sample variance S. Also, let y = (y. Y2 ..yp m, Ya) be a data set with a sample mean ỹ and a sample variance S. Finally, let z = x - y such that z- (X1- yı, X2 – y2 -- .X - Ya... . , Xn – Yn). Show that i -i - ỹ.
- 2) The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (0, 0+1), where is the true but unknown voltage of the circuit. Suppose that Y₁, ..., Yn denotes a random sample of such readings. a) Show that Y is a biased estimator of 0, and compute the bias. b) Find a function of Y that is an unbiased estimator of 0. 2) Sunnogo that the random vario obcorvation tributionLet ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.1. Let X1, X,.., X, random sample from distribution be a a Geometric 2....., fx (x) = p(1– p)*| F, (y,) = ? x-1 x=1,2,3,.... Fy, (y,) = ? P(Y, <1) =? а. b. С.
- 3 a. Let X have a Poisson distribution with a mean of 3. Define Y = (X-3) ^2 , specify pdf from Y.b. Let Y have a Poisson distribution with a mean of 5. Define Y=Z+X where Z and X are independent, where X is a random variable at point a, determine the distribution and pdf of Z.Let X~ N(0, 1) and let Y = -X. (a) Show that Y~ N(0, 1) (by showing that Y has cdf Þ). (b) What is the distribution of X + Y? (c) Are X and Y independent?Let X1, X2,..Xn be a random sample from a Poisson distribution with variance 0. Find the UMVUE for Pe(X=0). R(Ctrl) -
- If X has an F distribution with ν1 and ν2 degrees offreedom, show that Y = 1Xhas an F distribution with ν2 and ν1 degrees of freedom.What does the memoryless property refer to? An event cannot occur if it has already occured within a recent specified time period. The distribution of the time until an event does not depend on how much time has already passed. The occurrence of an event is independent of the number of times that event has already occurred. None of the above. Suppose X~ GAM (a, 3). Which of the following statements are true? Select all that apply. 3 is a location parameter. E[X]=a3 a is a location parameter. 3 is a shape parameter. 3 is a scale parameter. a is a shape parameter. a is a scale parameter. For a > 1, the probability density function (pdf) is monotonic decreasing. For a < 1, the probability density function (pdf) is monotonic decreasing. None of the above.Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.