The text from the image is as follows:2. Let $ X \sim U[0,1] $ and $ Y_n = \frac{n-1}{n} X $. Prove that $ Y_n \xrightarrow{P} X $.This exercise involves probability theory and requires proving the convergence in probability of a sequence of random variables $ Y_n $ to a random variable $ X $. Here, $ X $ follows a uniform distribution over the interval [0, 1]. The sequence $ Y_n $ is defined as a scaled version of $ X $. The task is to show that as $ n $ approaches infinity, $ Y_n $ converges in probability to $ X $.

Given that, X~U0,1 Yn=n-1nX Consider, ltn→∞PYn-X≥ε=ltn→∞Pn-1nX-X≥ε =ltn→∞P-1nX≥ε =ltn→∞PX≥nε =ltn→∞[1-nε] =0 Hence, the condition is satisfied. Therefore, Yn→XP

Answered: 2. Let X ~ U[0, 1] and Yn = X. Prove…

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2. Let X ~ U[0, 1] and Yn = X. Prove that Y, 4x.

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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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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The text from the image is as follows:

2. Let $ X \sim U[0,1] $ and $ Y_n = \frac{n-1}{n} X $. Prove that $ Y_n \xrightarrow{P} X $.

This exercise involves probability theory and requires proving the convergence in probability of a sequence of random variables $ Y_n $ to a random variable $ X $. Here, $ X $ follows a uniform distribution over the interval [0, 1]. The sequence $ Y_n $ is defined as a scaled version of $ X $. The task is to show that as $ n $ approaches infinity, $ Y_n $ converges in probability to $ X $.

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