6.9
Uniform Distribution
The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.
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The uniform distribution is a continuous probability distribution associated with events that are equally likely to occur.
Its probability density is expressed by a rectangular function where 'a' and 'b' are the lower and upper cut-offs, respectively.
For example, the voltage provided by the electricity company is uniformly distributed, say, between 122 and 126 volts.
In this case, the probability density is plotted as a function of the voltage supplied.
The total area under the graph should always be one. Since the range is 4 volts, the height must be one divided by 4.
One might wonder what is the probability of any household getting a voltage less than 123 volts?
It can be found from the area under the segment, which is the product of the width and height of the section.
This mean voltage supplied is the sum of the cut-off values divided by two, which in this case is a 124 volts.
The standard deviation is given by the range divided by the square root of twelve, which is found to be 1.2 volts.
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Q1: What is a uniform distribution and how is it represented?
A uniform distribution is a continuous probability distribution where all events have equal likelihood of occurring. It is represented by a rectangular function with lower cut-off 'a' and upper cut-off 'b'. The probability density function creates a rectangular shape on a graph, where the total area under the curve always equals one.
Q2: How do you calculate the probability of an event in a uniform distribution?
In a uniform distribution, probability equals the area under the curve segment for that event. Calculate this by multiplying the width of the segment by its height. For example, if voltage ranges from 122 to 126 volts with height 0.25, the probability of voltage below 123 volts is 1 volt × 0.25, or 0.25.
Q3: What is the formula for the mean of a uniform distribution?
The mean of a uniform distribution is calculated by adding the lower cut-off 'a' and upper cut-off 'b', then dividing by two: μ = (a + b) / 2. For example, if voltage ranges from 122 to 126 volts, the mean is (122 + 126) / 2, which equals 124 volts.
Q4: How is the standard deviation calculated for a uniform distribution?
Standard deviation in a uniform distribution equals the range divided by the square root of twelve: σ = (b - a) / √12. For a 4-volt range, the standard deviation is 4 / √12, or approximately 1.2 volts. This measures how spread out the values are within the distribution.
Q5: Why must the total area under a uniform distribution curve equal one?
The area under the probability density curve represents the total probability of all possible outcomes. Since one of the outcomes must occur with certainty, the total probability equals one. In a uniform distribution, this rectangular area is calculated as width (range) multiplied by height (probability density).
Q6: What does it mean when data follows a uniform distribution between two values?
When data follows a uniform distribution between values 'a' and 'b', every outcome within that range has an equal probability of occurring. For instance, baby smiling times uniformly distributed between 0 and 23 seconds means any duration in that range is equally likely, with mean 11.5 seconds and standard deviation 6.64 seconds.
Q7: How does a histogram from sample data relate to a theoretical uniform distribution?
A histogram constructed from sample data creates an empirical distribution that closely matches the theoretical uniform distribution when the data truly follows uniform distribution rules. This visual comparison helps verify whether observed data aligns with the expected rectangular shape and equal probability assumptions of the uniform model.