Find k ,such that the function
`P(x)={{:(k({:(4),(x):}),,x=0","1","2","3","4,kgt0),(0,"otherwise."):}`
is a probability mass function (p.m.f.)

Explanation

`P(x) ={{:(k((4)/(x)), x = 0","1","2","3","4, k gt 0),(0,"otherwise"):}`
`therefore P[X = 0] = k{:((4),(5)):} =k xx ""^(4)C_(0) = k(1) = k`
`P[X = 1] = k{:((4),(1)):} = k xx ""^(4)C_(1) = k (4) = 4k`
`P[x = 2] = k((4),(2)) = k xx ""^(4)C_(2) = k((4xx3)/(1xx2)) = 6k `
`P [x = 3] = k((4),(3)) = k xx ""^(4)C_(3) = k(4) = 4k`
`P[X =4] = k((4),(4)) = k xx ""^(4)C_(4) = k (1) = k`
Since `P[x = x] ` is p.m.f., `sum_(x=0)^(4) P[X = x] = 1`
` therefore P[X = 0 ] + P [X = 1] +P[X = 2] +P[X = 3] + P[x = 4] = 1`
`therefore k + 4k + 6k + k = 1`
`therefore 16k = 1 " "therefore k = (1)/(16)`