Dermine the value of c so that fx satisfies the condition of

Dermine the value of c so that f(x) satisfies the condition of being a p.m.f. for the random variable x.

f(x) = c(x^2-3x^3)   x=0,1,2...,n

Solution

summation of f(x) = c (x² - 3x³) = 1

Summation of x² = n (n+1) (2n+1) / 6

Summation of 3x³ = 3[ n(n+1) /2 ]²

Net summation =

[  n (n+1) (2n+1) / 6 ] - 3[ n(n+1) /2 ]²

On simplifying this, we get

[ -9n⁴ - 14n³ - 3n² + 2n / 12 ] * c = 1

So,

c = -12 / (9n⁴ + 14n³ + 3n² - 2n)

Hope this helps

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