b)Let X to be a random sample from a distribution with a probability density function given byf(x; 0) = {0x0-¹,if 0 < x < 1, 0 € {1,2},elsewhereOIt is desired to test a null hypothesis Ho: 0 = 1 against the alternative hypothesis H₁:0 = 2.Suppose that the test rejects Ho if x ≥ii) Calculate the power of the test.

Answered: O Let X to be a random sample from a…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The probability density function of the random variable X is given by 3. f(2) = , 0 Sz<2 otherwise 8 Calculate the variance Var(X). Enter your answer numerically and accurate to 2 decimal places. For example, if the value you calculated is 0.998, you should enter 1.00; if the value you calculated is 0.941, you should enter 0.94.
For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?
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Suppose that X is a random variable having the following probability density function given by (2(0-x) x>0 f(x) = elsewhere Find the value of c such that an interval from x to cx is a (1-a)100% confidence interval for the parameter 8.
b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
The error involved in making a certain measurement is a continuous random variable X with cumulative distribution function 0, for x 2. Calculate the probability that the error in measurement has an absolute value of less than one.
For the random variable X with the probability function below, answer the following questions. f(x)= { 3x^2 , 0<x<1 { 0, otherwise a. Find the cdf of X. b. Find the probability that X is less than 0.5. c. Find E(X).
Suppose the lifespan (in months) of a smartphone battery can be modeled as a continuous random variable with CDF F(x) = 1 − e-x/3 x ≥ 0 What is the probability that the battery lasts between 12 to 15 months?
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6. Suppose that the random variables X and Y have joint probability density function given by x+y, 0

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O Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0xª- S0x0-1, if 0