Question

The discrete random variable X has the following probability function.
P(X = x) = `{{:("k"x,  x = 2","  4","  6),("k"(x - 2),  x = 8),(0,  "otherwise"):}`
where k is a constant. Show that k = `1/18`

Answer 1

From the data

P(x = 2) = kx

= 2k

P(x = 4) = kx

= 4k

P(x = 6) = kx

= 6k

P(x = 8) = k(x – 2)

= k(8 – 2) = 6k

Since P(X = x) is a probability mass function

`sum_(x = 2)^8` P(X = x) = 1

`sum_("i" = 2)^oo` P(xi) = 1

i.e P(x = 2) + P(x = 4) + P(x = 6) + P(x = 8) = 1

2k + 4k + 6k + 6k = 1

18k = 1

∴ k = `1/18`