# 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))# ?

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#k=2/(n(n+1))#

##### 1 Answer

See the proof below.

#### Explanation:

The sum of the probabilities must be equal to

#sumP(X=r)=sum_(r=1)^nkr=1#

Therefore,

#ksum_(r=1)^nr=1#

We know that the sum of the first

#sum_(r=1)^n r=n/2(n+1)#

So,

#k*n/2(n+1)=1#

#k=2/(n(n+1))#