Variance and standard deviation

Learning Objectives Define a continuous random variable and its probability density function (PDF). Calculate the expected value (mean) of a continuous random variable using definite integrals. Define variance and standard deviation as measures of spread for a continuous probability distribution. Calculate the variance of a continuous random variable using the computational formula involving integrals. Calculate the standard deviation by taking the square root of the variance. Interpret the standard deviation as a measure of the typical distance from the mean for a continuous variable. How can we measure the 'spread' of something that can take on an infinite number of values, like the exact time a bus will arrive? 🚌 Let's use the power of continuous functions...

TermDefinitionExample Continuous Random VariableA variable that can take on any value within a given interval or range. Unlike discrete variables, there are infinitely many possible values.The exact height of a student, which could be 175cm, 175.1cm, 175.11cm, etc. Probability Density Function (PDF)A continuous function, f(x), that describes the relative likelihood of a continuous random variable. The total area under the curve of a PDF over its entire domain must equal 1.A function f(x) = 1/10 for 0 ≤ x ≤ 10, and f(x) = 0 otherwise. This describes a variable with an equal chance of being any value between 0 and 10. Expected Value (Mean, μ)The weighted average of all possible values of a continuous random variable, representing its central tendency or 'center of mass'. It is cal...

Expected Value (Mean) of a Continuous Random Variable E(X) = \mu = \int_{-\infty}^{\infty} x \cdot f(x) \,dx To find the mean, you integrate the product of the variable 'x' and its probability density function f(x) over the entire domain where f(x) is non-zero. Variance of a Continuous Random Variable (Computational Formula) Var(X) = \sigma^2 = E(X^2) - [E(X)]^2 = \left( \int_{-\infty}^{\infty} x^2 \cdot f(x) \,dx \right) - \mu^2 This is the most common way to calculate variance. First, find the expected value of X-squared by integrating x²f(x). Then, subtract the square of the mean (μ) that you calculated previously. Standard Deviation \sigma = \sqrt{Var(X)} The standard deviation is simply the positive square root of the variance. This returns the measure...