A random variable X that assumes the values x1, x2, …, xk is called a discrete uniform random va

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A random variable X that assumes the values x1, x2, …, xk is...

A random variable $X$ that assumes the values $x_{1}, x_{2}, \ldots, x_{k}$ is called a discrete uniform random variable if its probability mass function is $f(x)=\frac{1}{k}$ for all of $x_{1}, x_{2}, \ldots, x_{k}$ and 0 otherwise. Find the mean and variance of $X$

Key Concepts

Random Variable

A random variable is a function that assigns numerical values to the outcomes of a random experiment, providing a quantitative measure of the outcomes. It allows the transition from abstract outcomes to numerical analysis through associated probabilities.

Discrete Uniform Distribution

A discrete uniform distribution is a probability distribution where a finite number of outcomes are equally likely to occur. Every outcome in the sample space has the same probability, which simplifies analysis and calculations related to probabilities, expectations, and variances.

Probability Mass Function

The probability mass function (PMF) is a function that maps each outcome of a discrete random variable to its probability of occurrence. For a discrete uniform distribution, the PMF assigns an equal constant value to all outcomes, reflecting the idea that every outcome is equally likely.

Mean (Expected Value)

The mean or expected value of a random variable is a measure of central tendency, calculated as the weighted sum of all possible values of the variable, with weights given by their respective probabilities. In the context of a discrete uniform distribution, each value contributes equally to the mean.

Variance

Variance is a measure of the spread or dispersion of a set of values in a probability distribution. It quantifies how much the outcomes of a random variable differ from its mean. For a discrete uniform distribution, the variance is calculated by taking the expected value of the squared deviation of the variable from its mean, reflecting the uniform spread of outcomes around the average.

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