Random Variables Presentation
| Introduction to Random Variables | ||
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| • A random variable is a variable whose value is determined by the outcome of a random experiment. | ||
| • It represents a numerical quantity associated with a random event or experiment. | ||
| • Random variables can be discrete or continuous, depending on the nature of the outcomes. | ||
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| Discrete Random Variables | ||
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| • Discrete random variables take on a finite or countably infinite set of possible values. | ||
| • Examples include the number of heads obtained in multiple coin flips or the number of red cards drawn from a deck. | ||
| • Probability mass function (PMF) is used to describe the probability distribution of a discrete random variable. | ||
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| Continuous Random Variables | ||
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| • Continuous random variables can take on any value within a specified range or interval. | ||
| • Examples include the height of individuals in a population or the time taken to complete a task. | ||
| • Probability density function (PDF) is used to describe the probability distribution of a continuous random variable. | ||
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| Probability Distribution Functions | ||
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| • Probability distribution functions describe the likelihood of different values occurring for a random variable. | ||
| • For a discrete random variable, the PMF provides the probabilities for each possible value. | ||
| • For a continuous random variable, the PDF represents the relative likelihood of different intervals. | ||
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| Expected Value | ||
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| • The expected value, also known as the mean or average, is a measure of the central tendency of a random variable. | ||
| • It represents the average value that would be obtained over a large number of experiments or trials. | ||
| • The expected value is calculated by summing the product of each possible value and its corresponding probability. | ||
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| Variance and Standard Deviation | ||
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| • Variance measures the spread or dispersion of a random variable around its expected value. | ||
| • It quantifies the average squared deviation from the mean. | ||
| • Standard deviation is the square root of the variance and provides a measure of the typical deviation from the mean. | ||
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| Joint Probability Distribution | ||
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| • In some cases, we may have multiple random variables that are related to each other. | ||
| • The joint probability distribution describes the probabilities of different combinations of values for these variables. | ||
| • It can be represented using a joint probability mass function (for discrete variables) or a joint probability density function (for continuous variables). | ||
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| Marginal Probability Distribution | ||
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| • The marginal probability distribution focuses on a single random variable from a joint probability distribution. | ||
| • It provides the probabilities of different values for that specific variable, ignoring the values of other variables. | ||
| • Marginal probability distributions can be obtained by summing or integrating over the other variables. | ||
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| Conditional Probability Distribution | ||
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| • Conditional probability distribution describes the probability distribution of one variable given the value of another variable. | ||
| • It provides information about how the distribution of one variable changes when the other variable takes on a specific value. | ||
| • Conditional probability distributions can be obtained by dividing the joint probability distribution by the marginal probability distribution. | ||
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| Applications of Random Variables | ||
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| • Random variables are widely used in various fields, including statistics, economics, engineering, and finance. | ||
| • They help in modeling and understanding uncertain events and forecasting outcomes. | ||
| • Random variables provide a mathematical framework for analyzing and making decisions based on probabilities. | ||
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