Question

Cast a die a number of independent times until a six appears on the up side of the die. (a) Find the \(\operatorname{pmf} p(x)\) of \(X\), the number of casts needed to obtain that first six. (b) Show that \(\sum_{x=1}^{\infty} p(x)=1\). (c) Determine \(P(X=1,3,5,7, \ldots)\). (d) Find the \(\operatorname{cdf} F(x)=P(X \leq x)\).

Short answer

The pmf is \(p(x) = [(5/6)]^{x-1}(1/6)\), the sum of all these probabilities equals 1. The probability that the first six occurs on an odd number trial can be determined similar to step 2 (just for odd numbers). The cdf is given by \(F(x) = 1 - (5/6)^x\).

Step-by-step solution

Determine the pmf

The probability that the first six occurs on the \(x\)th roll is given by the formula for the probability mass function (pmf) of a geometric distribution, \(p(x) = (1-p)^{x-1}p\), where \(p\) is the probability of success on a single trial. In this case, the 'success' is rolling a six, and \(p = 1/6\). Therefore, \(p(x) = [(5/6)]^{x-1}(1/6)\).

Summation of the pmf from x=1 to infinity

To validate this pmf, it's necessary to show that the sum of probabilities over all possibilities is equal to 1. We have to prove \(\sum_{x=1}^{\infty} p(x)=1\). Essentially this is a geometric series with common ratio \(5/6\), less than 1. Therefore, the series will converge and the sum should equal \(1/6 / (1-5/6) = 1\).

Calculate the probability for odd rolls to find the first 6

To find \(P(X=1,3,5,7, \ldots)\), sum the probabilities of the first 6 appearing at an odd number of trials. This comes down to summing the geometric series over odd indices, which can be calculated similarly to step 2, just with alteration of the start and end point.

Determine the cdf F(x)

The cumulative distribution function (cdf) of \(X\), is \(F(x) = P(X \leq x)\). This is equivalent to the probability that you get the first six in \(x\) or fewer rolls. To calculate the cdf we sum the probability of getting a 6 on each roll from 1 to \(x\). Therefore, we sum the pmf from 1 to \(x\). This arithmetic sum will lead to \(F(x) = 1 - (5/6)^x\).

Key concepts

Understanding Probability Mass Function (PMF)

The probability mass function, commonly abbreviated as PMF, is a function that gives the probability that a discrete random variable is exactly equal to some value. In the provided exercise, the discrete random variable is the number of times we cast a die until we roll a six.

The formula for the PMF in this geometric distribution is given by \( p(x) = (1-p)^{x-1}p \) where \( p \) represents the probability of rolling a six in a single cast, which is \( \frac{1}{6} \). So, the PMF for rolling a six on the \( x \)th cast is \( p(x) = \left(\frac{5}{6}\right)^{x-1}\frac{1}{6} \).

PMF is a foundational concept in understanding distributions as it directly measures the likelihood of specific outcomes. It's key in plotting distribution graphs and making predictions about experiments or processes.

Summing Up with Series Convergence

Series convergence is an essential concept in calculus and analysis that determines if the sum of an infinite sequence of numbers has a finite limit. In our context, we ascertain the sum of probabilities for all possible outcomes to ensure this aligns with the fundamental property of probabilities summing to one.

In the second step of the solution, we come across the infinite series \( \sum_{x=1}^{\infty} p(x) \), which is a geometric series due to the consistent ratio between successive terms. The ratio here is \( \frac{5}{6} \), which is less than 1, ensuring that the sum of the infinite series converges to a finite value, specifically to 1, confirming the validity of our probability distribution.

Cumulative Distribution Function (CDF)

The cumulative distribution function, or CDF, is a fundamental concept in probability theory and statistics representing the probability that a real-valued random variable \( X \) will take a value less than or equal to \( x \). In simpler terms, it tells us the chance of an event happening at or before a certain point.

For our geometrically distributed random variable, the CDF is calculated by summing up the PMF from the first cast up to the \( x \)th cast: \( F(x) = P(X \leq x) = 1 - \left(\frac{5}{6}\right)^x \). This function gives us a running total of probabilities, making it easier to understand the distribution of outcomes over a given number of trials.

Geometric Series in Probability

A geometric series is a series with a constant ratio between successive terms, greatly relevant in probability, especially when dealing with the geometric distribution like in our die-casting scenario.

When summing the probabilities of rolling a first six on an odd number of throws, we're working with a geometric series. To find \( P(X=1,3,5,7, \ldots) \), we adjust our series to include only the odd indices. Despite this modification, the principles guaranteeing series convergence remain applicable, and the sum provides the total probability of such an event occurring.