An online calculator that calculates the mean, standard deviation and probability of a continuous uniform probability distribution is presented. A second calculator that calculates \( x_1 \) (inverse problem) such that \( P(X \lt x_1) = p \) given \(p \) is also included.

Continuous Uniform Probability Distribution

A continuous uniform probability distribution has the probability density function of the form
\[f(x) = \begin{cases}
\dfrac{1}{b-a} \quad \text{for} \quad a \le x \le b \\
\\
0 \quad \text{for} \quad x \lt a \quad \text{or} \quad x \gt b \\
\end{cases}
\]
and whose graph is shown below.

The probability that the random variable \( X \) is less than \( x_1 \) is given by
\[ \displaystyle P(X \lt x_1) = \int_{a}^{x_1} \dfrac{1}{b-a} \; dx \]
The mean, variance and standard deviation of a continuous uniform probability distribution, as defined above, are given by:
Mean = \( \dfrac{1}{2}(a +b) \)
Variance = \( \dfrac{(b-a)^2}{12} \)
Standard Deviation = \( \sqrt{\dfrac{(b-a)^2}{12}} \)
We present two calculators.

1 - Find the mean, standard deviation and probability \( P(X \lt x_1) \) given \( a , b \) and \( x_1 \)

\( a \) = , \( b \) = ,
\( x_1 \) =

Decimal Places =

Output

2 - Inverse Problem: Find \( x_1 \) such that \( P(X \lt x_1) = p \) given \( a , b \) and \( p \):