Show that $$f(y)=\begin{cases} 2y+\frac32&0<y<\frac12\\ 0&\text{otherwise} \end{cases}$$ is a probability density function and determine the expected value of the corresponding random variable $Y$.

What are the steps to complete this exercise?


closed as off-topic by 5xum, TheGeekGreek, Parcly Taxel, drhab, Dave Jul 18 '17 at 12:54

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You have to show: $f \ge 0$ on $\mathbb R$, $f$ is integrable and $\int_{ - \infty}^{\infty}f(y)dy=1$.

The expected value is given by $E(Y)=\int_{ - \infty}^{\infty}yf(y)dy$.


Step 1: Identify the two properties of a probability density function for an absolutely continuous random variable.

$$\forall y\in\Bbb R~(f(y)\geqslant 0)\\ \int_\Bbb R f(y)\operatorname d y = 1$$

Step 2: Show that this function has them.

Step 3: Apply the definition of expectation for such a random variable.

$$\Bbb E(Y) = \int_\Bbb R y~f(y)\operatorname d y$$


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