Uniform distribution corresponds to experiments where all the outcomes are equally likely.

Consider the case of rolling a fair dice which can take any of the 6 possible values and all the values are equally likely and the probability of any of the values is going to be 1/6 or in other words, the random variable can only take the value 1/6

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We call such a random variable as the uniform random variable and its distribution is a uniform distribution

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Similarly, we have other such experiments where all the outcomes are equally likely and we have a uniform random variable

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In general, we can talk of the support of the random variable(the values that the random variable can take as to lie between two numbers say ‘a’ and ‘b’)

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whereas for the uniform random variable, we have the probability as the following:

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The term in the denominator ‘b-a+1’ gives the number of elements between ‘a’ and ‘b’ as per the counting principle

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The first thing we need to prove is that the probability of any value that the random variable can take is greater than equal to 0

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We can prove this by considering the equation for this distribution

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So, as long as ‘b’ is greater than ‘a’ which will be the case as we are defining the range from ‘a’ to ‘b’, ‘b-a+1’ would be greater than or equal to 0 and hence this quantity is always going to be positive.

The other question is if the sum of probability values for all the values that the random variable can take is 1:

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The range of k would be from a to b instead of 1 to infinite
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This again we can answer by taking all the terms in the formula:

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Each of the values in the summation has the same probability value and the total number of terms is given as ‘b-a+1

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As Uniform distribution satisfies the properties of a standard PMF, we can say that it’s a valid distribution.

References: PadhAI

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