Continuous Uniform Distributions

Calculating Probabilities Above/Below Values

Learning Objectives

Calculate the probability of $x$ being at least, more than, at most or less than a value.

Probability of Exactly One Value

For continuous distributions:

The probability of being exactly at one value is zero.
Ie: $P(x= X)=0$

Because of this:

$P(x\ge X)=P(x= X)+P(x\gt X) = 0+P(x\gt X) = P(x\gt X)$
$P(x\le X)= P(x= X)+P(x\lt X) = 0+P(x\lt X) = P(x\lt X)$

At Least or More Than

For the probability of at least or more than $X$ , ie: $P(x\ge X)$ or $P(x\gt X)$

$x_1 = X$ (the lowest value in the $x$ -range)
$x_2 = b$ (the highest possible $x$ -value)

This gives $P(x_1 \le x \le x_2) = \frac{x_2 - x_1}{b-a} = \frac{b-X}{b-a}$ .

At Most or Less Than

For the probability of at most or less than $X$ , ie: $P(x\le X)$ or $P(x\lt X)$ :

$x_1 = a$ (the lowest possible $x$ -value)
$x_2 = X$ (the highest value in the $x$ -range)

This gives $P(x_1 \le x \le x_2) = \frac{x_2 - x_1}{b-a} = \frac{X-a}{b-a}$

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An Introduction to Business Statistics for Analytics (1st Edition) Copyright © 2024 by Amy Goldlist; Charles Chan; Leslie Major; Michael Johnson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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