Triangular Distribution
Abstract
Triangular Distribution
| The triangular distribution is a statistic method to represent a continuous probability distribution in cases where the available sample data is limited but the correlation between variables is known. Adhering to the fact that it is used in situations where data is scarce it is also known as “lack of knowledge” distribution. Its parameters consist of a lower bound, upper bound and mode(value that has the highest or best chance of occurring in a given set of values). |  |
Uses
Managing Projects : Triangular distributions in parallel with PERT distribution is used relied upon for project management to model events taking place within a specified interval.
Business Simulations : Triangular distributions have an extensive use in simulating business decisions. When information regarding the distribution of an outcome is vague(limited to its smallest and largest values and most likely outcome), triangular helps simulate the outcome which is used heavily in corporate financing.
Introduction
Triangular distribution uses the following parameters :
| Parameter | Description | Constraint |
|---|---|---|
| a | Lower Limit | a ≤ c |
| b | Upper Limit | b ≥ c |
| c | Mode | c ∈ [a , b] |
Triangular Distribution helps us simulate events with minimal sample data in the following way. First we gather the parameters.
Minimum Value : If the value is time ,we take the best time, if the value is a quantity, we take the least quantity and so on.
Maximum Value : If the value is time ,we take the worst time, if the value is a quantity, we take the least quantity and so on.
Mode : This would signify a value which is estimated to have the best chance of occurring.
Using these three parameters, we can apply the two continuous linear functions(pdf) which generate a simulation the events that can occur by generating a triangle. The probability of a value occurring is signified by the height of the triangle at that point, with the mode having the maximum height.
Formulae
The triangular distribution is a continuous distribution defined on the range x ∈ [a , b] with (“Triangular Distribution,” n.d.)
Probability distribution function :
Cumulative Distribution Function :
Mean :
Variance :
Real-Life Application
Monte Carlo Simulation
When faced with significant uncertainty in making a forecast or estimate, some methods replace the uncertain variable with a single average number. The Monte Carlo Simulation instead uses multiple values and then averages the results. The most common method used to chart out the Monte Carlo simulation is the triangular distribution, which is to say that the Monte Carlo method is essentially a triangular distribution.
Monte Carlo simulations have a vast array of applications in fields that are plagued by random variables, notably business and investing. They are used to estimate the probability of cost overruns in large projects and the likelihood that an asset price will move in a certain way. (Fairchild, Misra, and Shi 2016)
This in itself is used in multiple real life scenarios to this day :
Pricing stock options : The potential price movements of the underlying asset are tracked given every possible variable. The results are averaged and then discounted to the asset’s current price. This is intended to indicate the probable payoff of the options
Portfolio valuation : A number of alternative portfolios can be tested using the Monte Carlo simulation in order to arrive at a measure of their comparative risk
Fixed income investments : The short rate is the random variable here. The simulation is used to calculate the probable impact of movements in the short rate on fixed rate investments.
Problems and Solutions
When Josh arrives to class, he is typically 4 minutes late but he could arrive on time(best case) or be up to 10 minutes late.
a. What is the probability that he arrives no later than 2 minutes late?
b. What is the probability that he is over 7.5 minutes late?
Thus, the probability of that he arrives no later than 2 minutes late is 0.1 and the probability that he is over 7.5 minutes late is 0.1042 .
In a short survey on the length of YouTube advertisements, it is found that the most likely length of the advert is 17 seconds. YouTube policy states that the length of an advertisement is at least 4 seconds and almost 45 seconds. Calculate the probability that 2 randomly selected advertisements are both longer than 35 seconds.
Thus, the probability of two consecutive adverts being greater than 35 seconds is 0.00759 .
Conclusion
This report highlights one of the most unique and simplest methods to estimate occurrences even with a severe scarcity of data. Triangular distribution is something that can be used in everyday situations(where we know the best and worst cases and have a estimate of a “frequent occurrence” by laymen with a basic knowledge of its workings and the formula to find area of a right angled triangle.
The versatility of this distribution lies in it simplicity, speed of inferring data from it and the fact that it operates with very little data and works well with estimates about values(This helps in scenarios where we might not be aware of the exact value for parameters) which in especially useful in scenarios like predicting stock values, and scheduling projects where we can only estimate a close figure for maximum, minimum and mode.