Suppose that Y possesses the density function a Find the

Step 1 of 5

(a)

Suppose that Y has a probability density function,

\(f(y)=\left\{\begin{array}{ll} c y, & 0 \leq y \leq 2, \\ 0, & \text { elsewhere. } \end{array}\right.\)

We need to find the value of c that makes f(y) a probability density function.

If  f(y) is a density function for a continuous random variable, then

\(\int_{-\infty}^{\infty} f(y) d y=1\) ……….(1)

Hence we can calculate the value of k using equation (1),

\(\begin{array}{l} \int_{0}^{2} c y d y=1\\ \int_{0}^{2} y d y=\frac{1}{c}\\ \int_{0}^{2} y d y=\left.\left(\frac{y^{2}}{2}\right)\right|_{0} ^{2}=2\\ \frac{1}{c}=2, c=\frac{1}{2} \end{array}\)

Hence the value of \(c=\frac{1}{2}\) that makes f(y) a probability density function.