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

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Answer 1 · 2017-10-18T05:51:32

See the proof below.

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

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

Therefore,

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

We know that the sum of the first $n$ counting numbers is $n/2(n+1):$

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

So,

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

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

$QED$