# The probability distribution function of a discrete variable X is given by: P(X=r) = kr, r=1,2,3,...,n, where k is a constant. Show that k=2/(n(n+1)) ?

## $k = \frac{2}{n \left(n + 1\right)}$

Oct 18, 2017

See the proof below.

#### Explanation:

The sum of the probabilities must be equal to $1 :$

$\sum P \left(X = r\right) = {\sum}_{r = 1}^{n} k r = 1$

Therefore,

$k {\sum}_{r = 1}^{n} r = 1$

We know that the sum of the first $n$ counting numbers is $\frac{n}{2} \left(n + 1\right) :$

${\sum}_{r = 1}^{n} r = \frac{n}{2} \left(n + 1\right)$

So,

$k \cdot \frac{n}{2} \left(n + 1\right) = 1$

$k = \frac{2}{n \left(n + 1\right)}$

$Q E D$