AMTH 108 Chapter 3 Notes

Discrete Random Variables

1. Random Variable Definition

• A Random Variable is defined as a function that assigns specific real numbers to events in a sample space. The notation for this is:

  • $X: F \rightarrow \mathbb{R}$
    
    Where:

    • F is a subcollection of events in a sample space E.

    • ℝ denotes the set of all real numbers.

• The constraint for Random Variables ensures that for every real number x, the set $E_x = {o \mid X(o) \leq x}$ is an event, which aligns with the probability framework.

2. Discrete Set

• A set S is termed discrete if it can be matched one-to-one with a subset of the integers, meaning the elements in S can be counted or enumerated (countable).

• Categories of discrete sets include:

  • Finite Sets: Countable with a finite number of elements.

    • Examples: Even integers, Odd integers, integers divisible by a positive integer, Multiples of 2π.

  • Countably Infinite Sets: These sets have an infinite number of counts, such as the set of all integers, which can be listed but not exhausted.

3. Functions - Basics

• A function defines a relationship between two sets, D (domain) and R (range):

  • $f: D \rightarrow R$

• Key Properties of Functions:

  • Each element in D must correspond to exactly one element in R (multi-valued functions are disallowed).

  • The image of D under f, denoted as ${f(x) \mid x \in D}$, is referred to as Im(f).

  • Functions can yield the same output for different inputs: $f(d1) = f(d2)$ for $d1 \neq d2$ .

4. One-to-One and Onto Functions

• A function is considered one-to-one (injective) if:

• $f(x1) = f(x2)$ implies $x1 = x2$, ensuring unique mapping for distinct inputs.

• A function is termed onto (surjective) if every element of the range is covered by at least one element from the domain. If the image of the function Im(f) equals R, it is referred to as onto.

• If a function is both one-to-one and onto, it is termed bijective, indicating it possesses an inverse function denoted as $f^{-1}$, thus facilitating a two-way correspondence.

5. Useful Summations

• Key summations that are crucial in discrete mathematics and probability include:

  • $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$, which provides the sum of the first n integers.

  • $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$, yielding the sum of the squares of the first n integers.

  • $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$, representing the sum of the cubes of the first n integers.

6. Geometric Series

• Finite Geometric Series:

  • The sum of terms in a geometric sequence $r^k$ for $k = 0, 1, …, N$ can be represented as:
    
    $\sum_{k=0}^{N} r^k = \frac{1 - r^{N+1}}{1 - r}, r \neq 1$.

• Infinite Geometric Series: When the absolute value of r is less than 1, the sum is given by:

  • $\sum_{k=0}^{\infty} r^k = \frac{1}{1 - r}$

7. Random Variables - Basics

7.1 Definitions of Random Variables

• Discrete Random Variable:

  • A discrete random variable can take on a countably finite or infinite number of values, such as the outcomes of rolling dice which can yield discrete results (1 through 6).

• Characteristics of Random Variables include:

  • Random variables can partition a sample space (the entire universe of possible outcomes), where $U = \bigcup<em>{x} E</em>x$, and each event $E<em>x$ is distinct such that $E</em>x \cap E_y = \emptyset$ if $x \neq y$.

8. Probability Function & Examples of Random Variables

• Function P quantifies events, yielding probabilities that fall within the interval [0,1], representing the likelihood of different outcomes.

8.1 Rolling Two Dice Example

• When rolling a red die and a blue die, the possible outputs can be represented as ordered pairs $(r, b)$. The sum random variable is defined as:

  • X:

  • $X(r, b) = r + b$

  • A probability table for totals from the two dice is constructed, providing probability values from $P(X = 2)$ to $P(X = 12)$ based on various unique outcomes of the roll.

9. Height Example

• Random variable H quantifies height, influenced by a myriad of genetic factors and environmental conditions.

• While H can be identical for two individuals, it defaults to a function, ensuring the principle that each individual yields a unique height measurement.

10. Doughnuts Example

• Random Variable X(·) represents the number of doughnuts sold, which varies based on sales fluctuations observed over multiple days.

• Various statistical methods, including regression analysis and confidence intervals, are utilized to assess the accuracy and reliability of sales forecasts.

11. Voltage Measurements Example

• Thermal noise is a phenomenon indicating that voltage measurements are inherently random, influenced by various physical factors including temperature and resistance changes.

12. Discrete Random Variables Defined

• Discrete Random Variables are characterized by their ability to produce countable outcomes, distinguishing them from continuous variables, which can take any value in a range.

12.1 Output Size Examples

• Example of discrete outputs includes height measured with finite precision versus a continuous variable where increments can be infinitely small.

13. Density Functions

13.1 Definition of Discrete Density Function

• The density function $f_X$ defined for a discrete random variable illustrates:

  • $f(x) = P[X = x]$, establishing the probability associated with specific outcomes.

13.2 Examples & Properties

• Outcomes from the Rolling Dice can be illustrated through the density function, showcasing the likelihood of each potential sum after rolling two dice.

14. Cumulative Distribution Function (CDF)

14.1 Construction and Properties

• The CDF, denoted as $F(x)$, is defined through:

  • $F(x) = P[X \leq x]$, which is computed from the outcomes produced by the density function.

14.2 Graphical Representation

• The CDF graph represents a step function, with jumps occurring at each non-trivial value, signifying probabilities accumulating as x increases.

15. Expectation (Mean) of a Random Variable

15.1 Expected Value Definition

• The expected value of a random variable is computed using the formula:

  • $E[X] = \sum_{w \in W} wP[X = w]$, aggregating values weighted by their probabilities.

15.2 Variance and Standard Deviation

• Variance is expressed as $Var(X) = E[(X - \mu)^2]$, capturing the degree of deviation from the mean:

  • The standard deviation, denoted as $\sigma = \sqrt{Var(X)}$, provides a measure of the spread in the distribution of the random variable's values.

16. Moment Generating Functions

16.1 Definition and Use

• The Moment Generating Function, represented as $m_X(t) = E[e^{tX}]$, encapsulates the moments of the distribution and is a central tool in probability theory for analyzing random variables.

16.2 Recovery of Moments from Moment Generator

• Moments can be recovered from the moment-generating function using:

  • $\mu<em>n = E[X^n] = m^{(n)}</em>X(0)$, illustrating the relationship between the moments and the generating function.

17. Popular Discrete Distributions

17.1 Discrete Uniform Distribution

• The characteristics of Discrete Uniform Distribution are detailed here, with emphasis on its properties, mean, and variance calculations relevant to applications.

17.2 Bernoulli Trials, Geometric, and Binomial Distribution

• This includes a comprehensive analysis of mean, variance, and application scenarios pertinent to these distributions, especially in scenarios concerning success and failure.

17.3 Poisson Distribution

• The Poisson Distribution is discussed with practical implications, showcasing its application in real-world contexts, particularly in events that occur randomly over a fixed interval.