Uniform Probability Distribution Calculator

Uniform Probability Distribution Calculator

Calculation Results:

Minimum Value (a):

Maximum Value (b):

Mean (E[X]):

Variance (Var[X]):

Standard Deviation (SD[X]):

Specific Value (x):

Probability Density Function (f(x)):

Cumulative Distribution Function (F(x)):

Enter a Specific Value (x) to calculate PDF and CDF.

Understanding the Uniform Probability Distribution

The uniform probability distribution is a fundamental concept in statistics and probability theory. It describes a scenario where all outcomes within a given interval are equally likely. Unlike distributions like the normal distribution, which has a peak, the uniform distribution maintains a constant probability density across its entire range.

What is a Uniform Distribution?

A continuous uniform distribution is defined over an interval [a, b], where 'a' is the minimum value and 'b' is the maximum value. For any value 'x' within this interval, the probability density function (PDF) is constant. Outside this interval, the probability density is zero.

Key Characteristics and Formulas:

• Probability Density Function (PDF), f(x):

  For a continuous uniform distribution over the interval [a, b], the PDF is given by:

  f(x) = 1 / (b - a) for a ≤ x ≤ b

  f(x) = 0 otherwise

  This means that the height of the distribution is constant across the interval, and its area sums to 1 (as required for any probability distribution).

• Cumulative Distribution Function (CDF), F(x):

  The CDF gives the probability that a random variable X takes a value less than or equal to x. For a uniform distribution:

  F(x) = 0 for x < a

  F(x) = (x - a) / (b - a) for a ≤ x ≤ b

  F(x) = 1 for x > b

  The CDF starts at 0, increases linearly to 1 across the interval [a, b], and then stays at 1.

• Mean (Expected Value), E[X]:

  The mean represents the average value of the distribution. For a uniform distribution, it's simply the midpoint of the interval:

  E[X] = (a + b) / 2

• Variance, Var[X]:

  Variance measures the spread of the distribution. For a uniform distribution:

  Var[X] = (b - a)² / 12

• Standard Deviation, SD[X]:

  The standard deviation is the square root of the variance, providing a more interpretable measure of spread in the same units as the data:

  SD[X] = √((b - a)² / 12)

When to Use a Uniform Distribution?

The uniform distribution is often used when:

• There is no prior information suggesting that some outcomes are more likely than others within a given range.
• Modeling random processes where outcomes are truly random and evenly spread, such as the outcome of a fair die roll (discrete uniform) or a random number generator (continuous uniform).
• As a baseline or null hypothesis in statistical tests.

Example Scenario:

Imagine a bus arrives at a stop every 10 minutes, but you don't know the exact schedule. You arrive at the bus stop at a random time. The waiting time for the bus can be modeled as a uniform distribution between 0 minutes (if you arrive just as the bus does) and 10 minutes (if you just missed the bus). In this case, a = 0 and b = 10.

• Minimum Value (a): 0 minutes
• Maximum Value (b): 10 minutes
• Mean Waiting Time: (0 + 10) / 2 = 5 minutes. On average, you'd wait 5 minutes.
• Probability Density (f(x)): 1 / (10 – 0) = 0.1. This means for any specific minute within the 0-10 range, the "density" of probability is 0.1.
• Probability of waiting less than 3 minutes (F(3)): (3 – 0) / (10 – 0) = 0.3. There's a 30% chance you'll wait less than 3 minutes.

Use the calculator above to explore different minimum and maximum values, and see how the mean, variance, and probability density change!