# The probability Distribution Function Formula

The Probability Density Function(PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. The probability Density function is defined by the formula,

 P(a

## Questions on the probability distribution function

### Question 1:

The pdf of a distribution is given as $f(x)= \left\{\begin{matrix}x;\; for\ 0< x< 1 \\ 2-x;\; for \ 1< x< 2 \\ 0;\; for\ x> 2 \end{matrix}\right \}$

Calculate the density within the interval $(0.5< x< 1.5)$

### Solution:

$P(0.5< x< 1.5)=\int_{0.5}^{1.5}f(x)dx$ $=\int_{0.5}^{1}f(x)dx+\int_{1}^{1.5}f(x)dx$ $=\int_{0.5}^{1}xdx+\int_{1}^{1.5}(2-x)dx$ $=\left ( \frac{x^{2}}{2} \right )_{0.5}^{1}+\left ( (2x-\frac{x^{2}}{2}) \right )_{1}^{1.5}$

= 3/4

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