Question

question 5 let x be the number of students who attend an office hours session a day prior to an exam. x is known to be distributed as follows. p(x)=kx 0≤x≤5 what is the value of k? 0 1/15 0 0 0 5 5/15

Explanation:

Step1: Recall probability - mass function property

For a discrete probability - mass function $p(x)$, $\sum_{x}p(x)=1$. Here, $x = 0,1,2,3,4,5$ and $p(x)=kx$ for $x = 0,1,2,3,4,5$. So, $\sum_{x = 0}^{5}kx=1$.

Step2: Expand the sum

$\sum_{x = 0}^{5}kx=k\sum_{x = 0}^{5}x$. We know that $\sum_{i = 1}^{n}i=\frac{n(n + 1)}{2}$, and $\sum_{x = 0}^{5}x=0 + 1+2+3+4+5=\frac{5\times(5 + 1)}{2}=15$.

Step3: Solve for k

Since $k\sum_{x = 0}^{5}x=1$ and $\sum_{x = 0}^{5}x = 15$, then $k\times15=1$. So, $k=\frac{1}{15}$.

Answer:

$\frac{1}{15}$