Question

. 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

The probability mass function (pmf) of the smallest observation from a sample of size five from a discrete distribution where each point from 1 to 6 (inclusive) has a probability of 1/6, is determined by \[ g_1(y_1) = \left(\frac{7-y_{1}}{6}\right)^5 - \left(\frac{6-y_{1}}{6}\right)^5 \], where y_1=1,2,...6. The pmf is zero at all other points. The formulas for continuous distribution are not applicable as they consider a continuous spectrum of outcomes as opposed to a countable set of outcomes presented in this exercise.

Step-by-step solution

Assumptions and Definitions

All six points have the same probability of 1/6. An event where the smallest number is \(y_1\) means all five numbers are at least \(y_1\). So, that probability would be \(\left(\frac{7-y_{1}}{6}\right)^5\), since there are \((7 - y_1)\) possible points and each has a probability of \(1/6\), so this event has a cumulative distribution function (CDF) of \[\left(\frac{7-y_{1}}{6}\right)^5\]. The probability mass function (pmf) is the derivative of the cumulative distribution function (CDF), which for discrete variables means difference between \(y_{1}\) and \(y_{1}-1\).

Deriving pmf for Y1

By applying the formula for pmf as the differential of cdf for discrete variables, find the difference between the probabilities for \(y_1\) and \(y_1 - 1\), as G(Y1) - G(Y1-1). This is given as \[\left(\frac{7-y_{1}}{6}\right)^5 - \left(\frac{6-y_{1}}{6}\right)^5\]

Result

The pmf of the smallest observation of a random sample of size 5 from the given distribution is \(g_1(y_1) = \left(\frac{7-y_{1}}{6}\right)^5 - \left(\frac{6-y_{1}}{6}\right)^5\), where \(y_1=1,2,...6\), and it is zero elsewhere.

Justify non-applicability of continuous distributions

Discrete distribution is used if the number of outcomes can be counted in a set, whereas continuous distribution is used if the set of possible outcomes can take on values in a continuous range. The smallest observation from a sample in a discrete distribution cannot be modeled with the formulas derived under the assumption of a continuous distribution. Since we are dealing with discrete numbers (the outcomes of a dice roll), it doesn't make sense to use a continuous distribution model which assumes a continuous spectrum of possible outcomes.

Key concepts

Discrete Distribution

Understanding discrete distribution is crucial for grasping various concepts in probability theory. A discrete distribution is characterized by a set of distinct possible outcomes, each with its own probability. These outcomes are countable, like the results of rolling a die or flipping a coin.

In the given exercise, the distribution of the die's outcomes is discrete since there are six countable and separate results (1 through 6), each with a probability mass function (pmf) value of \(\frac{1}{6}\). This uniform distribution provides the foundation from which we can explore more complex probability scenarios, such as determining the pmf of the smallest value in a random sample.

Using discrete distributions, we can construct probability mass functions to represent the likelihood of each outcome. The pmf is a fundamental component in probability, as it tells us exactly how likely it is to obtain each possible value from our distribution. For instance, if you were to roll a fair die, the pmf would give a probability of \(\frac{1}{6}\) to each outcome, depicting the fair nature of the die.

It's important to remember that the sum of all the probabilities for every possible outcome in a discrete distribution must equal 1, reflecting the certainty that one of the outcomes will occur.

Cumulative Distribution Function

The cumulative distribution function (CDF) is another significant concept in probability theory. Unlike the pmf, which gives the probability of a single outcome, the CDF tells us the probability of obtaining a value less than or equal to a certain point. It essentially accumulates probabilities and is defined for every number in the distribution's support.

For discrete distributions, the CDF is a step function that jumps up by the probability of each value in the distribution. In the exercise, to find the pmf of the smallest observation (\(y_1\)) in a random sample of size 5 from a uniform distribution, the CDF is initially considered. Since the outcomes are discrete, we determine the cumulative probability by raising the fraction representing the available outcomes above the smallest number to the power of the sample size, as shown by \[\left(\frac{7-y_{1}}{6}\right)^5\].

The relationship between the pmf and CDF in a discrete setting is intimate; the pmf can be viewed as the 'difference' between the CDF values for consecutive outcomes. Therefore, to get the pmf from the CDF, you can subtract the CDF value at one point from the CDF value at the next point. This step-by-step increase reflects the essence of discrete probability and shows how each outcome contributes to the overall distribution.

Random Sample

A random sample relates closely with statistical and probability analyses. It consists of randomly chosen elements from a larger set or distribution, which is meant to represent a 'miniature' of the entire set. The main property of such a sample is that each member of the larger set has an equal chance of being included in the sample. This ensures unbiased representation and allows for the application of probability theory to infer conclusions about the larger set from the sample.

In our exercise, a random sample of size 5 is drawn from the discrete uniform distribution of a die's outcomes. Each of these samples has an equally likely chance of being anything from 1 to 6. The task requires us to deduce the pmf of the smallest observed value in these random samples. This scenario illustrates a common practice in statistical studies, where the behavior of a sample reflects on the possible attributes of the population from which it's drawn.

Moreover, random sampling stresses the importance of each sample's independence and identical distribution—often referred to as i.i.d. These properties ensure that each sample gives new and unbiased information about the underlying population, making it possible to make accurate predictions and understandings about the population as a whole.