Question

If the p.m.f of a. r.v. Xis given by

Givenif x = 0, 1, 2, ........ 5 = 0,

0, otherwise,

then which of the following is not true?

Options

• P(X ≤ 2) = P(X ≥ 3)

• P(X ≤ 1) = P(X ≥ 4)

• P(X ≤ 2) ≥ P(X ≥ 4)

• P(X ≤ 3) ≤ (X ≥ 3)

Answer 1

P(X ≤ 3) ≤ (X ≥ 3)

Explanation:

Given, P(X = x) = `("^5C_x)/2^5`

X = x	0	1	2	3	4	5
P(X)	`("^5C_0)/2^5`	`("^5C_1)/2^5`	`("^5C_2)/2^5`	`("^5C_3)/2^5`	`("^5C_4)/2^5`	`("^5C_5)/2^5`

Here, ⁵C₀ = ⁵C₅, ⁵C₁ = ⁵C₄, ⁵C₂, ⁵C₃

∴ P(X ≤ 3) = `1/2^5 [1 + 5 + 10 + 10] = 26/2^5`

P(X ≥ 3) = `1/2^5 (10 + 5 + 1) = 16/2^5`

∴ P(X ≤ 3) > P(X ≥ 3)

∴ Option (d) is not true.