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I was reading the uniform random variable from sheldon ross

it is given that X is a uniform random variable on the interval $(\alpha,\beta)$ if the probability density function of X is given by

$f(x)=\left\{  \frac{1}{\beta-\alpha}\, if \, \alpha \lt x\lt \beta \\ 0\, otherwise\right\}$

Since F(a)=$\int_{- \infty}^{a}f(x)dx$, it follows from above that the distribution function of a uniform random variable on the interval

$(\alpha,\beta)$ is given by

$F(a)=\left\{ 0 \, a \leq \alpha \\ \frac{a-\alpha}{\beta-\alpha} \; \alpha \lt a\lt \beta\\1 \,\, a \geq \beta\right\}$

My query is how they derived the distribution function from the density function?

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Distribution function or CDF is just probability before a given point. So its the integral of density function(in case of continuous distribution) from -$\infty$ to the given point. In uniform distribution all the probability lies between two points $\alpha$ and $\beta$ in a square shape. So obviously if you calculate probability before $\alpha$, it'll be 0 and 1 before $\beta$.
If you want to find out CDF of a point between $\alpha$ and $\beta$, you'll need to take integral with $\alpha$ as lower bound and the given point as upper bound.

 

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  1. The probability distribution function / probability function has ambiguous definition. They may be referred to:
    • Probability density function (PDF)
    • Cumulative distribution function (CDF)
    • or probability mass function (PMF) (statement from Wikipedia)
  2. But what confirm is:
    • Discrete case: Probability Mass Function (PMF)
    • Continuous case: Probability Density Function (PDF)
    • Both cases: Cumulative distribution function (CDF)

https://math.stackexchange.com/questions/175850/difference-between-probability-density-function-and-probability-distribution

http://www.igidr.ac.in/faculty/susant/TEACHING/CITI/sl_citi-02.pdf

 

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