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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 31E

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5.5. Suppose that X has mean m and variance o
(a) Prove Cantelli's inequality
P[X-mza]s
av0.

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3. If X is x² (23), find the following: a) P(14.85 < X < 32.01) b) constants a and b such that P(a < X < b) = 0.95 and P(X
(5) If Xis a r.v. with mean u and variance o?, using Chebyshev's-Inequality, AX-2 20) S (a) 0.5 (b) 0.05 (c)0.657 (d) 0.25.
3
Suppose that f (x) = x / 8 for 3 < x < 5. Determine the mean and variance of x.
1
6) Suppose the m.g.f. of the r.v. X is given below: MCA) = (uM/ R0) +(45 |no) exp (t) A(20/1201eapreb a) Find the mean and variance of X b) Find P(X>1)
8
5. If X is normally distributed with a mean of 6 and a variance of 25, find a) P(1.2 ≤X -2) c) P(6 ≤X ≤ 14) d) P(-4 35) f) P(|X-6 < 12.4) g) P(|X| < 7) h) Find c such that P(X ≥ c) = 0.1. And what is the percentile name of c? Find c such that P(|X - 6| < c) = 0.95 i) j) Find E(e0.3x).
i.i.d. Suppose X1,..., X, ' Uniform[0, 6] with unknown 0. Consider the following three estima- tors for 0: = max X,..., X, n + 1 max X1, .. Xm n n + 2 max X1, ... , Xn n + 1 Use the R code "Week 4 - compare estimators for uniform.R" in Canvas/Files/Discussion to conduct simulations to compare the three estimators in terms of their bias, standard error (SE), and mean squared error (MSE). In the simulation, suppose the true 0 = 1. (a, Simulate 10000 data sets with n = 10. Compute the bias, SE and MSE of each of the three estimators from the simulation. Present your simulation result in the following table. Then answer: Which one has the smallest bias? Which one has the smallest SE? Which one has the smallest MSE? 3 bias SE MSE 02 Table 1: Performance measure with n = 10 (b) pute the bias, SE and MSE of each of the three estimators from the simulation. Present Simulate 10000 data sets for each of the following n: n = 10, 20, 50, 100. Com- your simulation result in the following table.…
8.11.) Prove that X, the mean of a random sample of size n from a distribution that is N(0, ¹), -∞0 0, an efficient estimator of 0.
2 If X₁ and X₂ are the means of independent ran- dom samples of sizes n₁ and n₂ from a normal population with the mean u and the variance o2, show that the vari- ance of the unbiased estimator 2 w X₁ + (1-w). X₂ is a minimum when @ = n1 n₁ + n₂ idnu
20.5. Suppose that A, B, and C are positive, independent random variables with distribution function F. Show that the quadratic Az2+ Bz+ C has real zeros with probability foF(x²/4y) dF(x) dF(y).

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