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<p<b) = ab∫ f(p) dp

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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