Uniform Distribution Formula
A uniform distribution is a continuous probability distribution and relates to the events which are likely to occur equally. A uniform distribution is defined by two parameters, a and b, where a is the minimum value and b is the maximum value. It is generally denoted as u(a, b). The probability density function graphically is portrayed as a rectangle where \(b-a\) is the base and \(\frac {1}{b-a}\) is the height. Let us learn about the uniform distribution formula in more detail.
What Is Uniform Distribution Formula?
When the probability density function or probability distribution of a uniform distribution with a continuous random variable X is f(x)=1/b-a, then It can be denoted by U(a,b), where a and b are constants such that a<x<b. It is written as:
f(x) = 1/ (b-a) for a≤ x ≤b.
where,
- a is the minimum value
- b is the maximum value
In terms of mean μ and variance σ2, the probability density may be written as:
\(f(x)=\left\{\begin{array}{ll} \frac{1}{2 \sigma \sqrt{3}} & \text { for }-\sigma \sqrt{3} \leq x-\mu \leq \sigma \sqrt{3} \\ 0 & \text { otherwise } \end{array}\right.\)
Solved Examples Using Uniform Distribution Formula
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Example 1: For a uniform probability density function, the height of the function:
a. Is different for various values of x
b. Decreases as x increases
c. Is the same for each value of x
Solution:As the name suggests, a uniformly or symmetrical probability distribution of a finite continuous variable data series is called a uniform probability distribution function. The area under it is equal to 1 since it a flat probability density. The mean is equal to the median and all the values are equally probable.
Thus, for a uniform probability density function, the height of the function is the same for each value of x.
Answer: c. Is the same for each value of x
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Example 2: Using the uniform probability density function for random variable \(\mathrm{X}\), X ~ (0,23), find P(2<X<18).
Solution: \(P(2<X<20)=(20-2) \cdot \frac{1}{23-0}=\frac{18}{23}\)
Answer: P = 18/23
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