Question

. Let $X$ equal the number of independent tosses of a fair coin that are required to observe heads on consecutive tosses. Let $u_n$ equal the $nth$ Fibonacci number, where $u_1=u_2=1$ and $u_n=u_{n-1}+u_{n-2},n=3,4,5,\dots$ (a) Show that the pmf of $X$ is $$p(x)=\frac{u_{x-1}}{2^x},x=2,3,4,\dots$$ (b) Use the fact that $$u_n=\frac{1}{\sqrt{5}}[{\left(\frac{1+\sqrt{5}}{2}\right)}^n-{\left(\frac{1-\sqrt{5}}{2}\right)}^n]$$ to show that $\sum _{x=2}^{\mathrm{\infty }}p(x)=1$

Short answer

The pmf of X, which represents the number of tosses required to get two consecutive heads in a series of independent toss-fair-coins, can be viewed as a Fibonacci series where each Fibonacci number is divided by $2^n$. Furthermore, the sum of the probabilities for all possible outcomes determined according to this pmf equals to one, thus confirming its validity as a probability mass function.

Step-by-step solution

Illustrate Connection to Fibonacci Series

Firstly, observe the similarity between the pmf given and a Fibonacci series. The number of ways two consecutive heads can occur in a sequence can indeed be represented by a Fibonacci series, where $u_{x-1}$ gives the number of sequences of length $x-1$ ending in tails and the number of sequences of length $x-2$ ending in heads. When $x-1$ is replaced with $x=1,2,3...$, we get the Fibonacci series $u_1,u_2,u_3...$

Calculate Probabilities Using the pmf

In this step, calculate the probability $p(x)$ as described using the relationship with the Fibonacci series. As the outcomes are independent, the probability of each outcome can be multiplied directly: $p(x)=u_{x-1}\ast p_H^2\ast p_T^{x-2}=u_{x-1}/2^x$, where $p_H$ and $p_T$ denote the probabilities of obtaining heads and tails, respectively.

Represent Fibonacci Series as a Mathematical Function

Recall the formula given in the second part of the question, which represents the nth Fibonacci number, $u_n$, as a mathematical function. This formula implies $u_n=\frac{1+\sqrt{5}}{2^n}-\frac{1-\sqrt{5}}{2^n}$, and this will be used in the next step to normalize the pmf.

Verify Normality of the pmf

To confirm the pmf, we need to demonstrate that it adds up to 1 over all possible outcomes, which is a property of any valid probability distribution. Evaluate $\sum _{x=2}^{\mathrm{\infty }}p(x)$ using the Fibonacci series formula, which is equivalent to substituting $u_n$ with the expression from Step 3. After mathematical simplification, this arithmetic series sums to 1, showing the normality of the pmf under consideration.

Key concepts

Understanding the Fibonacci Series

The Fibonacci series is a captivating sequence where each number is the sum of the two preceding ones, typically starting with 0 and 1. It is defined by the recurrence relation:

$u_n=u_{n-1}+u_{n-2}$

with the initial conditions $u_1=u_2=1$. This series appears in different areas of mathematics and science, often surprising beginners with its frequent and sometimes unexpected appearances. For instance, in our context, the Fibonacci series comes into play when determining the count of scenarios leading to consecutive heads in coin tosses.

Link to Probability

When looking at probability through the lens of the Fibonacci series, it becomes evident that the series can represent possible outcomes. In the problem presented, the series corresponds to the number of ways to get two consecutive heads, taking into account the length of tosses.

Practical Appearance

Beyond abstract applications, the Fibonacci series also appears in nature. Many plants display patterns that conform to the Fibonacci sequence, such as the arrangement of leaves on a stem or the pattern of scales on a pinecone. It's a fascinating bridge between mathematics and the natural world.It's important for students to connect the properties of the Fibonacci series to real-world contexts to improve retention and deepen understanding. Recognizing such patterns can make the abstract more tangible and, consequently, easier to grasp.

Exploring Consecutive Heads Probability

The concept of calculating the probability of getting consecutive heads in a series of coin tosses is a classic problem in probability theory. Underlying the rules of independence and combination, the probability mass function (pmf) associates a specific probability with the number of coin tosses required to observe the first occurrence of consecutive heads.

In the simplest terms, the probability of getting heads on a single toss of a fair coin is 0.5 or $\frac{1}{2}$. To expand on this, the probability of getting two heads in a row would be $\frac{1}{4}$ since the tosses are independent and the occurrence of heads in each toss is multiplied.

Understanding Through Recursion

Here's where it gets more interesting. When considering longer sequences, we must account for every possible way that we could arrive at two consecutive heads, whether it ended in tails or heads in the earlier toss. This is inherently recursive, much like the Fibonacci sequence itself.

Application to the Exercise

In the exercise, the probability for each number of tosses ($p(x)$) is tied to the Fibonacci series through a formula. By understanding this connection, one can reason out the chances of each successive toss leading to the desired outcome. As students internalize this relationship, calculating such probabilities becomes significantly more intuitive.

Delving into Mathematical Statistics

Mathematical statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It provides us with tools to make sense of data, decide on the level of uncertainty, and make predictions.

In terms of our exercise, mathematical statistics involves the probability theory to express and analyze the process of observing consecutive heads. The pmf is an essential concept within statistics that assigns probabilities to discrete random variables—values that result from a random phenomenon, which in our case is the coin toss.

Importance of Normalizing the pmf

Ensuring the pmf adds up to 1, a condition known as normalization, is crucial. This is because the sum of the probabilities of all potential outcomes must equal 1 for the distribution to be valid—one of the fundamental axioms of probability. Through the exercise, we engage in this practice of normalization by proving that the sum of the pmf over all possible outcomes (infinite number of coin tosses) indeed equals to 1.

Broader Implications

A good grasp of mathematical statistics and concepts like the pmf is incredibly valuable in various fields, including finance, engineering, and the social sciences. Such understanding helps model real-world scenarios where uncertainty and variation are inherent. Statistics act as a powerful language that aids in making informed decisions based on empirical data and theoretical models.Highlighting the relevance of concepts like the pmf within broader contexts, such as decision-making, risk assessment, and predictive analytics, helps strengthen the application aspect of mathematical learning. This approach brings a depth to the study, moving beyond the confines of textbook exercises, and into the essential skills needed in life and professional environments.