Triangular Distribution Probability Calculator

The triangular distribution is commonly used in simulations, risk analysis, and probabilistic modeling. It is defined by a lower limit, upper limit, and a mode, representing the most likely value. Use this calculator to compute probabilities and visualize the distribution.

Lower Limit (a):

Upper Limit (b):

Mode (c):

Value for Random Variable (x):

P(X = ):

P(X < ):

P(X ≤ ):

P(X > ):

P(X ≥ ):

Mean:

Median:

Variance:

Understanding the Triangular Distribution

The Triangular Distribution is a continuous probability distribution defined by a lower limit, an upper limit, and a mode, which represents the most likely value. It is commonly used in simulations and risk analysis.

Key Components of the Triangular Distribution

Lower Limit (a): The minimum possible value.
Upper Limit (b): The maximum possible value.
Mode (c): The most likely value in the distribution.

Formula for Triangular Distribution

The probability density function (PDF) for the triangular distribution is given by:

\[ f(x|a,b,c) = \begin{cases} 0 & \text{for } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \leq x \leq c, \\ \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c \leq x \leq b, \\ 0 & \text{for } x > b \end{cases} \]

Step-by-Step Example

Let's assume the lower limit is \(a = 1\), the upper limit is \(b = 10\), and the mode is \(c = 5\). If we want to find the probability for \(x = 4\), we apply the formula above.

Mean, Median, and Variance

Mean (μ): \( \frac{a + b + c}{3} \)
Median: If \( c \leq \frac{a + b}{2} \), the median is \( a + \sqrt{(b-a)(c-a)/2} \). Otherwise, the median is \( b - \sqrt{(b-a)(b-c)/2} \).
Variance (σ²): \( \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18} \)

Triangular Distribution Probability Calculator￼