Question

A random variable X has the following probability distribution
(a) determine the value of a
(b) Find p ( 1 < x < 4 )
(c) P(1 ≤ x ≤ 4)
(d) Find P (x >2)
(e) Find the distribution function of x

Answer 1

(a) ∵ ∑P(X = x) = 1
∴ P(X = 0) + P(X = 1) + ..... + P(X = 4) = 1
⇒ 3a + 4a + 6a + 7a + 8a = 1
⇒28a = 1
⇒a = 1/28

(b) P(1 < x < 4) = P(X = 2) + P(X = 3)
= 6a + 7a = 13a = 13/28

(c) P(1 ≤ x < 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
= 4a + 6a + 7a + 8a
= 25a = 25/28

(d) P(X > 2) = P(X = 3) + P(X = 4)
= 7a + 8a = 15a = 15/28

(e) F(x) = 0 , -∞ < x < 0
= 3/28 , 0 ≤ x < 1
= (3/28 + 4/28) = 7/28 , 1 ≤ x < 2
= (3/28 + 4/28 + 6/28) = 13/28 , 2 ≤ x < 3
= (3/28 + 4/28 + 6/28 + 7/28) = 20/28 , 3 ≤ x < 4
= (3/28 + 4/28 + 6/28 + 7/28 + 8/28) = 1 , 4 ≤ x < ∞