Question

Let X be the amount of tune (in hours) the wait is to get a table at a restaurant. Suppose the cdf is represented by

F(x) = {0 x < 0
{x³/8 0 < x < 2
{1 x > 2

Use the cdf to determine P(X ≥ 1/2).
a. 0.5000
b. 0.0156
c. 0.9844
d. 0.0000
e. 1.0000
f. None of the above

Answer 1

To determine P(X ≥ 1/2) using the given CDF, calculate 1 - F(1/2). F(1/2) equals (1/2)³/8, which is 1/64. Thus, P(X ≥ 1/2) is 1 - (1/64), yielding the answer c. 0.9844.

Step-by-step explanation:

The question asks us to use the cumulative distribution function (CDF) to determine P(X ≥ 1/2). The CDF F(x) is given piecewise as follows: F(x) = 0 for x < 0, F(x) = x³/8 for 0 < x < 2, and F(x) = 1 for x > 2. To find P(X ≥ 1/2), we need to calculate 1 - P(X < 1/2), since P(X > x) for a continuous distribution is 1 - P(X < x).

Plugging x = 1/2 into the CDF, we get F(1/2) = (1/2)³/8 = 1/64. So, P(X < 1/2) = 1/64. Therefore, P(X ≥ 1/2) = 1 - (1/64) = 63/64, which is approximately 0.9844.

The correct answer to the question "Use the cdf to determine P(X ≥ 1/2)." is c. 0.9844.