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Suppose I have a continuous random variable modelled by a probability density function: $$f(x)=2x$$ Obviously the domain (x) needs to be restricted between parameters ##a## and ##b## such that ##\int_a^b f(x) \, dx = 1##

Setting ##a=0##, working out ##b## should prove straightforward: $$\int_0^b 2x \, dx = 1$$ $$ \big[x^2\big] _0^b = 1$$ $$b^2 - 0 = 1$$ $$b=1$$ So, the function is now: $$f(x) =\left\{ \begin{array}{l}

2x , & 0 \leq x \leq 1\\ 0 , & \textrm{otherwise} \end{array} \right. $$

Except that the area under this function obviously exceeds 1. Even taking ##f(x)## at the boundary, ##f(1)## returns a value of 2. What am I doing wrong? Why is something so simple not adding up?

Thanks