Verify that the functions are probability mass functions, and determine the requested probabilities.

f (x) = (3/4) (1/4) ^x, x = 0, 1, 2, ...

a. P (X = 2)

b. P (X ≤ 2)

c. P (X > 2)

d. P (X ≥ 1)

a) 3/64 = 0.046 (4.6%)

b) 63/64 = 0.9843 (98.43%)

c) 1/64 = 0.015 (1.5%)

d) 1/4 = 0.25 (25%)

Step-by-step explanation:

in order to verify that the f (x) is a probability mass function, then it should comply the requirement that the sum of probabilities over the entire space of x is equal to 1. Then

∑f (x) * Δx = 1

if f (x) = (3/4) (1/4) ^x, x = 0, 1, 2, ...

then Δx=1 and

∑f (x) = (3/4) ∑ (1/4) ^x = (3/4) * [ 1 / (1-1/4) ] = (3/4) * (4/3) = 1

then f represents a probability mass function

a) P (X = 2) = f (x=2) = (3/4) (1/4) ^2 = 3/64 = 0.046 (4.6%)

b) P (X ≤ 2) = ∑f (x) = f (x=0) + f (x=1) + f (x=2) = (3/4) + (3/4) (1/4) + 3/64 = 63/64 = 0.9843 (98.43%)

c) P (X > 2) = 1 - P (X ≤ 2) = 1 - 63/64 = 1/64 = 0.015 (1.5%)

d) P (X ≥ 1) = 1 - P (X < 1) = 1 - f (x=0) = 1 - 3/4 = 1/4 = 0.25 (25%)