Question

Let $X$ denote the number of hours you study during a randomly selected school day. The probability that $X$ can take the values of $x$, has the following form, where $k$ is some constant
$P(X=x)=\begin{array}{c}\{\begin{array}{cc}0,1& \text{if}x=0\\ Kx& \text{if}x=1\text{or}2\\ K(5-x)& \text{if}x=3\text{or}4\\ 0& \text{otherwise}\end{array}\end{array}$
find the value of $K$.

Answer 1

Given :

$P(X=x)=\begin{array}{c}\{\begin{array}{cc}0,1& \text{if}x=0\\ Kx& \text{if}x=1\text{or}2\\ K(5-x)& \text{if}x=3\text{or}4\\ 0& \text{otherwise}\end{array}\end{array}$

We know that $\sum P(X=x)=1$

$⟹P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X>4)=1$

$⟹0+Kx+Kx+K(5-x)+K(5-x)+0=1$

$⟹0+2\left(Kx\right)+2K(5-x)+0=1$

$⟹2K(x+5-x)=1$

$⟹2K\left(5\right)=1$

Hence, $K=\frac{1}{10}$