Identify the probability density function. Then find the mean, variance, and standard deviation

Name: Problem 14, Chapter 9: Calculus an Applied Approach
Uploaded: 2021-06-15T14:22:05-08:00
Duration: 2 min 11 s
Channel: Ahmad Reda
Description: Identify the probability density function. Then find the mean, variance, and standard deviation without integrating. f(x)=0.2,[0,5]

step 1 Prover 14 we want to identify this probability density function over this interval then find the

Identify the probability density function. Then find the...

Key Concepts

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the random variable. It represents the average distance of the data points from the mean, thereby offering an intuitive gauge of the variability in the uniform distribution.

Probability Density Function

A probability density function (pdf) describes the likelihood of a continuous random variable taking on a specific value. It must satisfy the properties that the function is non-negative for all possible values of the random variable and that the integral over its entire range equals 1. In this context, identifying the pdf involves recognizing that the function is defined and constant over its support.

Uniform Distribution

A uniform distribution is a type of probability distribution where every interval of the same length within the given range is equally likely to occur. For a continuous uniform distribution on the interval [a, b], the pdf has a constant value, which is 1/(b - a). This simplification allows for straightforward computation of the mean and variance without integration.

Mean

The mean (or expected value) of a continuous uniform distribution is calculated as the midpoint of the interval, given by (a + b)/2. This value represents the central or average outcome of the random variable and does not require integration due to the symmetry and constant nature of the distribution.

Variance

The variance of a continuous uniform distribution measures the spread of the distribution and is given by (b - a)²/12. This expression quantifies the average squared deviation from the mean, providing insight into the dispersion of the values in the distribution.

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