Mathematics | Probability Distributions Set 1 (Uniform Distribution)

1. Discrete Probability Distribution – If the probabilities are defined on a discrete random variable, one which can only take a discrete set of values, then the distribution is said to be a discrete probability distribution. For example, the event of rolling a die can be represented by a discrete random variable with the probability distribution being such that each event has a probability of [Tex]\:\frac{1}{6}[/Tex].
2. Continuous Probability Distribution – If the probabilities are defined on a continuous random variable, one which can take any value between two numbers, then the distribution is said to be a continuous probability distribution. For example, the temperature throughout a given day can be represented by a continuous random variable and the corresponding probability distribution is said to be continuous.

Uniform Probability Distribution –

• Example 1 – The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0, 25]. Find the formula for the probability density function [Tex]f(x)[/Tex] of the random variable [Tex]X[/Tex] representing the current. Calculate the mean, variance, and standard deviation of the distribution and find the cumulative distribution function [Tex]F(x)[/Tex].
• Solution – The first step is to find the probability density function. For a Uniform distribution, [Tex]f(x) = \frac{1}{b-a}[/Tex], where [Tex]b,\:a[/Tex] are the upper and lower limit respectively. [Tex] \therefore \[ f(x) = \begin{cases} \frac{1}{25-0} = 0.04, & 0\leq x\leq 25 \\ 0, & \text{otherwise} \\ \end{cases} \] [/Tex] The expected value, variance, and standard deviation are- [Tex] E(x) = \frac{b+a}{2} = \frac{25+0}{2} = 12.5 mA\\\\ V(X) = \frac{(b-a)^2}{12} = \frac{(25-0)^2}{12} = 52.08 mA^2\\\\ \text{Standard Deviation} = \sigma = \sqrt{V(x)} = \frac{25}{2\sqrt{3}} = 7.21 mA [/Tex] The cumulative distribution function is given as- [Tex] F(x) = \int \limits_{-\infty}^{x} f(x) dx [/Tex] There are three regions where the CDF can be defined, [Tex]x<0,\: 0\leq x\leq 25,\:25 < x[/Tex] [Tex] \[ F(x) = \begin{cases} 0, &x<0\\ \frac{x}{25}, &0\leq x\leq 25\\ 1, &25<x \end{cases} \] [/Tex]

References –