Random Variables PDF

Random Variable s In the realm of statistics and probability theory, random variables and probability distributions form the foundation for understanding and analyzing uncertain events. This presentation delves into the concepts of random variables, distinguishing between discrete and continuous ty...

Random Variable s In the realm of statistics and probability theory, random variables and probability distributions form the foundation for understanding and analyzing uncertain events. This presentation delves into the concepts of random variables, distinguishing between discrete and continuous types, and explores their associated probability distributions. By grasping these fundamental ideas, students will gain crucial insights into the mathematical representation of real-world phenomena, enabling them to model and predict outcomes in various fields, from physics to economics. Random Variables: The Building Blocks of Pro A random variable is a function that assigns a numerical value to each outcome of a random experiment or process. Typically denoted by uppercase letters (e.g., X, Y, Z), while it’s corresponding lowercase letter is used to represent one of its values. Random variables provide a mathematical framework for describing uncertain events. Random Variables: The Building Blocks of Pro Used in various fields such as statistics, physics, finance, and social sciences to model unpredictable phenomena. Typically denoted by uppercase letters (e.g., X, Y, Z), while it’s corresponding lowercase letter is used to represent one of its values. Random variables provide a mathematical framework for describing uncertain events. Random Variables: The Building Blocks of Proba Random Variables: The Building Blocks of Pro Definition Representation A random variable is a function that assigns a numerical Typically denoted by uppercase letters (e.g., X, Y, Z), value to each outcome of a random experiment or random variables provide a mathematical framework for process. describing uncertain events. Applications Types Used in various fields such as statistics, physics, finance, Classified into two main categories: discrete random and social sciences to model unpredictable phenomena. variables and continuous random variables, each with distinct properties and applications. Random variables serve as the cornerstone of probability theory, allowing us to quantify and analyze uncertainty in a systematic manner. By assigning numerical values to outcomes, they bridge the gap between abstract concepts and measurable quantities, facilitating statistical inference and decision-making in complex scenarios. Random Variables: The Building Blocks of Pro Classified into two main categories: discrete random variables and continuous random variables, each with distinct properties and applications. Discrete Random Variables: Counting the Possibilities 1 Definition A discrete random variable is a type of random variable that can only take on a countable number of distinct values. These values are typically whole numbers or integers, representing outcomes that can be counted or enumerated. 2 Characteristics Discrete random variables have a finite or countably infinite set of possible values. Each value corresponds to a specific probability, and the sum of all probabilities equals 1. 3 Examples Common examples include the number of heads in a series of coin tosses, the count of defective items in a production batch, or the number of customers arriving at a store in a given hour. 4 Probability Mass Function The probability distribution of a discrete random variable is described by its probability mass function (PMF), which assigns a probability to each possible value of the variable. Understanding discrete random variables is crucial for modeling and analyzing phenomena with distinct, countable outcomes. They form the basis for many statistical techniques and are widely used in fields such as quality control, inventory management, and demographic studies. Discrete Random Variables: Counting the Possibilities 1 Definition A discrete random variable is a type of random variable that can only take on a countable number(finite) of distinct values. These values are typically whole numbers or integers or nonnegative numbers , representing outcomes that can be counted or enumerated. 2 Characteristics Discrete random variables have a finite orpossible values. Each value corresponds to a specific probability, and the sum of all probabilities equals 1. Discrete Random Variables: Counting the Possibilities 3 Examples Common examples include the number of heads in a series of coin tosses, the count of defective items in a production batch, or the number of customers arriving at a store in a given hour. W= number of correct responses in a 20-item true or false test. X= number of randomly selected students to be interviewed by researcher. Y= number of randomly selected left-handed teachers in a faculty room Z= count of defective light bulbs among the 100 samples of light Discrete Random Variables: Counting the Possibilities Understanding discrete random variables is crucial for modeling and analyzing phenomena with distinct, countable outcomes. They form the basis for many statistical techniques and are widely used in fields such as quality control, inventory management, and demographic studies. Continuous Random Variables: Measuring the Unmeasurable Definition and Characteristics Properties and Examples Probability Density Function A continuous random variable is a type of Continuous random variables are The probability distribution of a continuous random variable that can take on any value characterized by their ability to take on any random variable is described by its within a given range or interval. Unlike value within their range, including fractional probability density function (PDF). Unlike discrete random variables, continuous or irrational numbers. Examples include the the probability mass function for discrete random variables have an uncountably exact time it takes to complete a task, the variables, the PDF represents the relative infinite number of possible values. They are precise weight of a manufactured product, likelihood of the variable falling within a used to model measurements on a or the exact amount of rainfall in a specific particular range of values. The area under continuous scale, such as time, distance, or location. The probability of a continuous the PDF curve over a specific interval gives temperature. random variable taking on any single exact the probability of the variable taking a value value is zero, as there are infinitely many within that interval. possible values. Continuous random variables provide a powerful tool for modeling real-world phenomena that involve measurements on a continuous scale. They are essential in fields such as physics, engineering, and finance, where precise measurements and continuous processes are common. Continuous Random Variables: Measuring the Unmeasurable Definition and Characteristics A continuous random variable is a type of random variable that can take on any value within a given range or interval. Unlike discrete random variables, continuous random variables have an uncountably infinite number of possible values. They are used to model measurements on a continuous scale, such as time, distance, or temperature.  Continuous Random Variables: Measuring the Unmeasurable Properties and Examples Continuous random variables are characterized by their ability to take on any value within their range, including fractional or irrational numbers. Examples include the exact time it takes to complete a task, the precise weight of a manufactured product, or the exact amount of rainfall in a specific location. The probability of a continuous random variable taking on any single exact value is zero, as there are infinitely many possible values.  Continuous Random Variables: Measuring the Unmeasurable Examples X= lengths of centimeters of shoes randonly selected high school students. Z= hourly temperature last sunday Y=heights of daisy plants in the backyard Continuous Random Variables: Measuring the Unmeasurable Continuous random variables provide a powerful tool for modeling real-world phenomena that involve measurements on a continuous scale. They are essential in fields such as physics, engineering, and finance, where precise measurements and continuous processes are common. Let’s have a game! Let’s have a drill !  Random Variables Determine if the random variable is discrete or continuous. a. W= number of points scored in the last season by randomly selected basketball players in the PBA b. X= heights in centimeters of randomly selected students inside the library. c. Y= count of birds in a nest d. Z= weights in kilograms of randomly selected dancers after taking up aerobics Probability Distributions When the outcomes in an experiment are mapped to their corresponding probabilities by means of table of values, graphs or equations, a probability distribution is formed. Like random variables, probability distributions can either be discrete or continuous. If RANDOM VARIABLE is discrete, its probability distribution is called a DISCRETE PROBABILITY DISTRIBUTION. Otherwise, its is called a CONTINUOUS PROBABILITY DISTRIBUTION. Discrete Probability Distributions: Mapping Countable Outcomes A discrete probability distribution is a mathematical function that describes the likelihood of each possible outcome for a discrete random variable. It provides a complete description of the probability of occurrence for all potential values of the variable. Discrete Probability Distributions: Mapping Countable Outcomes Discrete probability distributions are characterized by their probability mass function (PMF), which assigns a probability to each possible value. The sum of all probabilities in a discrete distribution must equal 1, reflecting the certainty that one of the outcomes will occur. Discrete Probability Distributions: Mapping Countable Outcomes Several well-known discrete probability distributions are widely used in various applications. These include the Binomial distribution (modeling the number of successes in fixed trials), the Poisson distribution (describing rare events in a fixed interval), and the Geometric distribution (representing the number of trials until the first success). Discrete Probability Distributions: Mapping Countable Outcomes Discrete probability distributions find applications in numerous fields, such as quality control (defect rates), reliability analysis (component failures), and game theory (strategic decision- making). They are essential tools for modeling and analyzing phenomena with countable outcomes. Discrete Probability Distributions: Mapping Countable Outcomes Applications: Understanding discrete probability distributions is crucial for making informed decisions and predictions in scenarios involving countable outcomes. These distributions provide a solid foundation for statistical inference and hypothesis testing in various real-world applications. Discrete Probability Distributions: Mapping Countable Outcomes Example 2: Consider the table. x 0 1 2 3 P(x) 0.2 0.3 0.3 0.2 Discrete Probability Distributions: Mapping Countable Outcomes Example 3: The given spinner Task: Find the is divided into four equal probability that sectors. the arrow will stop at 1, 2,3, 1 2 and 4 where the arrow will stop. Construct the 3 4 discrete probability of the random variable x. Discrete Probability Distributions: Mapping Countable Outcomes Example 4: The given spinner is divided into four equal sectors. 10 20 0 0 10 40 0 0 40 40 0 0 30 30 0 0 Discrete Probability Distributions: Mapping Countable Outcomes Example 5: When two fair dice are trown simultaneously and the numbers that come up are observed, the following are the possible outcomes: (1, 1) (1, 2) (1,3) (1,4) (1,5) (1,6) (2, 1) (2, 2) (2,3) (2,4) (2,5) (2,6) (3, 1) (3, 2) (3,3) (3,4) (3,5) (3,6) (4, 1) (4, 2) (4,3) (4,4) (4,5) (4,6) Discrete Probability Distributions: Mapping Countable Outcomes We define the random variable X as the sum of the two numbers as an outcome in throwing the two fair dice simultaneously. The possible values are 2, 3, 4,, 5, 6, 7, 8, 9, 10, 11, 12 from (1,1) to 1+1=2, (1,2) to 1+2=3. Discrete Probability Distributions: Mapping Countable Outcomes Example 6: A discrete random variable X has the following probability distribution: x 1 2 3 4 P(x) 0.1 ? 0.2 0.2 Discrete Probability Distributions: Mapping Countable Outcomes Example 7: A discrete random variable X has the following probability distribution: x 1 2 3 4 0.2 P(x) 0.21 29c 0.29 1