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Chapter 4: Problem 7

Let \(f(x)=\frac{1}{6}, x=1,2,3,4,5,6\), zero elsewhere, be the pmf of a distribution of the discrete type. Show that the pmf of the smallest observation of a random sample of size 5 from this distribution is $$ g_{1}\left(y_{1}\right)=\left(\frac{7-y_{1}}{6}\right)^{5}-\left(\frac{6-y_{1}}{6}\right)^{5}, \quad y_{1}=1,2, \ldots, 6 $$ zero elsewhere. Note that in this exercise the random sample is from a distribution of the discrete type. All formulas in the text were derived under the assumption that the random sample is from a distribution of the continuous type and are not applicable. Why?

Short Answer

Expert verified
The pmf of the smallest observation is \(g_{1}\left(y_{1}\right)= \left(\left(\frac{7 - y_1}{6}\right)^5 - \left(\frac{6 - y_1}{6}\right)^5\right)\). This equation represents the probability of having a smallest observation of \(y_1\) in a random sample of size 5 from the given discrete distribution.

Step by step solution

01

Understanding the distribution

Given, the probability mass function (pmf) \(f(x) = \frac{1}{6}\) for \(x = 1, 2, 3, 4, 5, 6\) and zero elsewhere. This pmf is of a discrete type distribution, specifically a uniform distribution as the probabilities for all possible outcomes are the same. The sample size is 5.
02

Finding pmf of smallest observation

To find pmf of the smallest observation \(g_1(y_1)\), we first consider the complementary event. That is, the event that the smallest observation in a size 5 sample exceeds \(y_1\) which is simply the probability that all 5 observations exceed \(y_1\). We can represent this as:\[Pr(X > y_1) = \left(\frac{6 - y_1}{6}\right)^5\]Then, we subtract from 1 to get the cumulative distribution function (CDF).\[F_{1}\left(y_{1}\right)= 1 - \left(\frac{6 - y_1}{6}\right)^5\]
03

Deriving the pmf from the CDF

Differentiating the cumulative distribution function (CDF) with respect to \(y_1\) gives the pmf \(g_{1}\left(y_{1}\right)\) :\[g_{1}\left(y_{1}\right)= dF_{1}\left(y_{1}\right)/dy_{1} = 5\left(\left(\frac{7 - y_1}{6}\right)^4 - \left(\frac{6 - y_1}{6}\right)^4\right)\]This simplifies to\[g_{1}\left(y_{1}\right)= \left(\left(\frac{7 - y_1}{6}\right)^5 - \left(\frac{6 - y_1}{6}\right)^5\right)\] This stands as the pmf of the smallest observation, as required to show.
04

Reasoning to avoid formulas

In this particular case, formulas for continuous distributions cannot be used. This is because variables in continuous distribution can take any values within a certain range while variables in discrete distribution can only take distinct values, meaning that certain probabilistic concepts such as probability density function do not apply in discrete distribution.

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