Question

Suppose the random variable X has a geometric distribution with p = 0.5, Determine the following probabilities:

(a) P(X=3) = _____
(b) P(X=6) = _____
(c) PX-8) = _____
(d) P(X≤2) = _____
(e) P(X>2) = _____
Round your answers to four decimal places (e.g. 98.7654)

Answer 1

The geometric distribution is a probability distribution that models the number of independent trials required until the first success is achieved. In this case, the random variable X has a geometric distribution with p = 0.5. The probabilities are: (a) P(X=3) = 0.125, (b) P(X=6) = 0.015625, (c) P(X>8) = 0.99609375, (d) P(X≤2) = 0.75, and (e) P(X>2) = 0.25.

Step-by-step explanation:

The geometric distribution is a probability distribution that models the number of independent trials required until the first success is achieved. In this case, the random variable X has a geometric distribution with p = 0.5.

To determine the probabilities:

• (a) P(X=3) = (1-p)^(3-1) * p = (1-0.5)^(3-1) * 0.5 = 0.125
• (b) P(X=6) = (1-p)^(6-1) * p = (1-0.5)^(6-1) * 0.5 = 0.015625
• (c) P(X>8) = 1 - P(X<=8) = 1 - (1-p)^8 = 1 - (1-0.5)^8 = 0.99609375
• (d) P(X≤2) = 1 - P(X>2) = 1 - (1-p)^2 = 1 - (1-0.5)^2 = 0.75
• (e) P(X>2) = 1 - P(X≤2) = 1 - 0.75 = 0.25