Mathematics | Probability Distributions Set 1 (Uniform Distribution)

Prerequisite – Random Variable

In probability theory and statistics, a probability distribution is a mathematical function that can be thought of as providing the probabilities of occurrence of different possible outcomes in an experiment. For instance, if the random variable X is used to denote the outcome of a coin toss (“the experiment”), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair).
Probability distributions are divided into two classes –

  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 :frac{1}{6}.
  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.

Cumulative Distribution Function –
Similar to the probability density function, the cumulative distribution function F(x) of a real-valued random variable X, or just distribution function of X evaluated at x, is the probability that X will take a value less than or equal to x.
For a discrete Random Variable,
 F(x) = P(Xleq x) = sum limits_{x_0leq x} P(x_0)
For a continuous Random Variable,
 F(x) = P(Xleq x) = int limits_{-infty}^{x} f(x)dx

Uniform Probability Distribution –

The Uniform Distribution, also known as the Rectangular Distribution, is a type of Continuous Probability Distribution.
It has a Continuous Random Variable X restricted to a finite interval [a,b] and it’s probability function f(x) has a constant density over this interval.
The Uniform probability distribution function is defined as-

     [ f(x) =  egin{cases} frac{1}{b-a}, & aleq x leq b\ 0, & 	ext{otherwise}\ end{cases} ]

Uniform Distribution graph

Expected or Mean Value – Using the basic definition of Expectation we get –

     egin{align*} E(x) &= int limits_{-infty}^{infty} xf(x) dx&\ &= int limits_{a}^{b} frac{x}{b-a} dx&\ &= frac{1}{b-a} int limits_{a}^{b} x dx&\ &= frac{1}{b-a} Big[ frac{x^2}{2}Big]_{a}^{b}&\ &= frac{b^2 - a^2}{2(b-a)}&\ &= frac{b + a}{2}&\ end{align*}

Variance- Using the formula for variance- V(X) = E(X^2) - (E(X))^2

     egin{align*} E(x^2) &= int limits_{-infty}^{infty} x^2f(x) dx&\ &= int limits_{a}^{b} frac{x^2}{b-a} dx&\ &= frac{1}{b-a} int limits_{a}^{b} x^2 dx&\ &= frac{1}{b-a} Big[ frac{x^3}{3}Big]_{a}^{b}&\ &= frac{b^3 - a^3}{3(b-a)}&\ &= frac{b^2 + a^2 + ab}{3}&\ end{align*}

Using this result we get –

     egin{align*} V(x) &= frac{b^2 + a^2 + ab}{3} - Big( frac{b+a}{2}Big) ^2 &\ &= frac{b^2 + a^2 + ab}{3} - frac{b^2+a^2+2ab}{4} &\ &= frac{4b^2 + 4a^2 + 4ab - 3b^2 - 3a^2 - 6ab}{12}&\ &= frac{(b-a)^2}{12}&\ end{align*}

Standard Deviation – By the basic definition of standard deviation,

     egin{align*} sigma &= sqrt{V(x)} \&= frac{b-a}{2sqrt{3}} end{align*}

  • 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 f(x) of the random variable X representing the current. Calculate the mean, variance, and standard deviation of the distribution and find the cumulative distribution function F(x).
  • Solution – The first step is to find the probability density function. For a Uniform distribution, f(x) = frac{1}{b-a}, where b,:a are the upper and lower limit respectively.

         	herefore  [  f(x) =  egin{cases}  frac{1}{25-0} = 0.04, & 0leq xleq 25 \  0, & 	ext{otherwise} \ end{cases} ]

    The expected value, variance, and standard deviation are-
     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\\ 	ext{Standard Deviation} = sigma = sqrt{V(x)} = frac{25}{2sqrt{3}} = 7.21 mA
    The cumulative distribution function is given as-
     F(x) = int limits_{-infty}^{x} f(x) dx
    There are three regions where the CDF can be defined, x<0,: 0leq xleq 25,:25 < x

         [ F(x) =  egin{cases} 0, &x<0\ frac{x}{25}, &0leq xleq 25\ 1, &25<x end{cases} ]

References –

Probability Distribution – Wikipedia
Uniform Probability Distribution – statelect.com

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This article is attributed to GeeksforGeeks.org

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