The random variable X has probability distribution P(X) of the following form, where k is some number: (a) Determine the value of k . (b) Find P(X < 2), P(X ≥ 2), P(X ≥ 2).
The given probability distribution P ( X ) for the random variable X is,
P ( X ) = { k , if x = 0 2 k , if x = 1 3 k , if x = 2 0 , otherwise
(a)
It is known that ∑ P ( X ) = 1 .
We know that ∑ P ( X ) is the sum of the probability distribution of the P ( X ) .
∑ P ( X ) = 1 k + 2 k + 3 k + 0 = 1 6 k = 1 k = 1 6
Thus, the value of k is 1 6 .
(b)
The probability for X < 2 , where X is a random variable.
P ( X < 2 ) = P ( X = 0 ) + P ( X = 1 ) = k + 2 k = 3 k = 3 × 1 6
Thus, the value of P ( X < 2 ) is 1 2 .
The probability for X ≤ 2 , where X is a random variable.
P ( X ≤ 2 ) = P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) = k + 2 k + 3 k = 6 k = 6 × 1 6
Thus, the value of P ( X ≤ 2 ) is 1.
The probability for X ≥ 2 , where X is a random variable.
P ( X ≥ 2 ) = P ( X = 2 ) + P ( X > 2 ) = 3 k + 0 = 3 × 1 6 = 1 2
Thus, the value of P ( X ≥ 2 ) is 1 2 .
The given probability distribution
(a)
It is known that
We know that
Thus, the value of k is
(b)
The probability for
Thus, the value of
The probability for
Thus, the value of
The probability for
Thus, the value of
