The random variable X has a probability distribution P(X) of the following form, where k is some arbitary real number:
`P(X=x)={{:(k",","if "x=0), (2k",","if "x=1), (3k",","if "x=2), (0",", otherwise):}`
(i) Determine the value of k `" "` (ii) Find `P(X lt 2), P(X le 2), P(X ge 2)`

Answer

N/a

Explanation

The probability distribution of `X` is

(a) Since `sumP(X)=1`,
Therefore `P(0)+P(1)+P(2)+P` (otherwise) `=1`
`impliesk+2k+3k+0=1implies6k=1impliesk=1/6`
(b) `P(Xlt2)=P(0)+P(1)-k+2k=3k=3xx1/6=1/2`
`P(Xle2)=P(0)+P(1)+P(2)=k+2k+3k=6k`
`=6xx1/2=1`
and `P(Xge2)=P(2)+P` (otherwise)
`=3k+0=3xx1/6=1/2`