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Chapter 5: Problem 2

Let \(Y_{1}\) denote the minimum of a random sample of size \(n\) from a distribution that has pdf \(f(x)=e^{-(x-\theta)}, \theta

Short Answer

Expert verified
The limiting distribution of \(Z_{n}\) is the exponential distribution with parameter 1.

Step by step solution

01

Determine the Cumulative Distribution Function (CDF) for \(Y_{1}\)

Start with the cumulative distribution function (CDF) for a single observation \(X\), which from given \(f(x)\) is \(F(x) = 1 - e^{-(x-\theta)}\) for \(x>\theta\). Since \(Y_{1}\) is the smallest value in a sample of size \(n\), its CDF will be the \(n\)th power of \(F(x)\), because this is the probability that \(n\) independent values \(X_i\) all exceed \(x\). So, \(F_{Y_{1}}(x) = (1 - e^{-(x-\theta)})^n\) for \(x>\theta\).
02

Calculate the Probability Density Function (PDF) for \(Y_{1}\)

Differentiate the CDF to find the probability density function (PDF) of \(Y_{1}\). By the chain rule, \(f_{Y_{1}}(x) = n e^{-(x-\theta)} (1 - e^{-(x-\theta)})^{n-1}\).
03

Change of Variables to Find the PDF of \(Z_{n}\)

Next, implement a change of variables to find the distribution of \(Z_{n} = n(Y_{1} - \theta)\) in terms of the distribution of \(Y_{1}\). The change of variable formula involves the Jacobian determinant of the transformation, but here the Jacobian is one, simplifying the computation. It turns out, the PDF of \(Z_{n}\) is \(f_{Z_{n}}(z) = e^{-z} (1 - e^{-z})^{n-1}\) for \(z>0\).
04

Limiting Distribution of \(Z_{n}\)

Use the Binomial Theorem to rewrite the PDF of \(Z_{n}\) and take the limit as \(n\) approaches infinity. The binomial expansion is \((1-e^{-z})^{n-1} = \sum_{k=0}^{n-1}(-1)^k \binom{n-1}{k} e^{-kz}\). But as \(n\) approaches infinity, only the \(k=0\) term retains its significance - all other terms vanish. Hence the limiting PDF is \(f(z) = e^{-z}\) for \(z>0\), which is the exponential distribution with parameter 1.

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Most popular questions from this chapter

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Hence, we can also say that \(\left\\{a_{n}\right\\}\) is a sequence of constant (degenerate) random variables. Let \(a\) be a real number. Show that \(a_{n} \rightarrow a\) is equivalent to \(a_{n} \stackrel{P}{\rightarrow} a\)

Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a distribution that is \(N\left(\mu, \sigma^{2}\right) .\) Find the limiting distribution of \(\bar{X}_{n}\).

Let the random variable \(Y_{n}\) have a distribution that is \(b(n, p)\). (a) Prove that \(Y_{n} / n\) converges in probability to \(p\). This result is one form of the weak law of large numbers. (b) Prove that \(1-Y_{n} / n\) converges in probability to \(1-p\). (c) Prove that \(\left(Y_{n} / n\right)\left(1-Y_{n} / n\right)\) converges in probability to \(p(1-p)\).

Let \(X\) be \(\chi^{2}(50)\). Approximate \(P(40

Let \(\bar{X}\) denote the mean of a random sample of size 100 from a distribution that is \(\chi^{2}(50)\). Compute an approximate value of \(P(49<\bar{X}<51)\).
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