A random variable X has a continuous uniform distribution if its probability density function f(x) is constant over the interval (α, β). The uniform distribution has a probability density function f(x) = k for α < x < β, where k is a constant, and is equal to 0 otherwise. All values from α to β are equally likely to occur, meaning the probability of X falling in any sub-interval of (α, β) is equal regardless of the interval's position within the range.

1. 1.8 Uniform Distribution

2. Rectangular or Uniform distributionA random variable X is said to have a continuous uniform distribution over aninterval (, ) if its probability density function is constant k over entire range of x.PROBABILITY DENSITY FUNCTIONf (x) = k,  &lt; X &lt;  = 0 otherwise

3. Rectangular or Uniform distributionThe uniform distribution, with parameters  and , has probability density function

4. Figure:Graph of uniform probability densityAll values of x from  to  are equally likely in the sense that the probability that x lies in an interval of width x entirely contained in the interval from  to  is equal to x/( - ), regardless of the exact location of the interval.Uniform distribution

5. Distribution function for uniform density functionUniform distribution

6. The Uniform DistributionMean of uniform distributionProof:

7. The Uniform DistributionVariance of uniform distributionProof:

8. The Uniform DistributionMoment generating function

9. Discrete Uniform distributionIf random variable assume finite no. of values with each value occuring with same probability Probability density function is f(x) = 1/n, X=x1,x2,…… xn