1. A stick of length L is broken at a random point uniformly distributed on the interval [0, L). Twonew sticks are formed.a) If X denotes the length of the longest new stick, calculate E(X) and V(X)b) If Y denotes the length of the shortest new stick, calculate E(Y) and V(Y) Hint: X +Y = Lc) An experiment consists on breaking sticks of length L independently as described before untilthe longest new stick has length of at least 0.9L. What is the expected number of sticks to bebroken in this experiment.

Here, we are given a stick of length L, which is broken into two parts. The stick is uniform on the…

Answered: A stick of length L is broken at a…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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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 tribution
20. Apart from the usual random errors of measurement, an instrument has one source of bias which cannot be totally eliminated by suitable adjustment. It is however possible to set the instrument so as to make the bias positive or negative. The instrument is used to take 2n independent measurements of a certain quantity; n of these measurements a = (x1, 82,- set for positive bias, and the other n measurements (y1, Y2, bias, so that ,In) are taken with the instrument Yn), taken with a setting of negative ... .... E(x;) = 01 + 02, E(y;)= 01 – 02, Var(x;) = Var(y;) = o? (a) Assuming X and Y are normally distributed, determine the maximum likelihood estimators of 01, 02 and o. (b) A random the measurements from the two instruments are give in the table below Obtain Positive bias X: | 24.58 23.98 24.42 25.6 27.22 26.05 Negative bias Y: 23.32 23.04 22.45 23.78 27.00 22.48 estimates for 01, 02 and o based on your estimators in (a).
a) Let X₁, X2, X3,..., Xn be a random sample of size n from population X. Suppose that X~N(0, 1) Σ=1 Xi and Y = -√n. √√n i) Show that the standard score of the sample mean X, is equal to Y.
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)). D
a) Let X₁, X₂, X3,..., Xn be a random sample of size n from population X. Suppose that X~N(0,1) and Y = -√n. Ei=1 Xi √n i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y² is between 0.02 and 5.02?
Let Xi NcO,1) and n =2 auel T=X,+ Xz Show that Tis a %3D Sufficieacy stanslic Sufficieney Statistic.
5. Let X1 - b(n1, p) and X2 ~ b(n2,p) be independent random variables. a) Find the m.g.f. of Y = X1+ X2. %3D b) How is Y distributed?
20. Apart from the usual random errors of measurement, an instrument has one source of bias which cannot be totally eliminated by suitable adjustment. It is however possible to set the instrument so as to make the bias positive or negative. The instrument is used to take 2n independent measurements of a certain quantity; n of these measurements x = set for positive bias, and the other n measurements (y1, Y2, . .. , Yn), taken with a setting of negative bias, so that (x1, x2, ..., xn) are taken with the instrument E(x;) = 01 + 02, E(y;) = 01 – 62, Var(x;) = Var(y;) : g² (a) Assuming X and Y are normally distributed, determine the maximum likelihood estimators of 01, 02 and o. (b) A random the measurements from the two instruments are give in the table below Obtain Positive bias X: 24.58 23.98 24.42 25.64 27.22 26.05 Negative bias Y: 23.32 23.04 22.45 23.78 27.00 22.48 estimates for 01, 02 and o based on your estimators in (a).

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