Question

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

{ 0 x < 0
F(x) = { x³/8 0 ≤ x ≤ 2
{ 1 x > 2
Use the cdf to determine P(X ≥ ½).
a) 0.5000
b) 0.0156
c) 0.9844
d) 0.0000
e) 1.0000
f) None of the above

Answer 1

To calculate P(X ≥ ½), we use the cumulative distribution function and the properties of continuous distributions. The CDF for X ≤ ½ is calculated and subtracted from one to find P(X ≥ ½), which is 63/64 or approximately 0.9844. Hence, the answer is c) 0.9844.

Step-by-step explanation:

The student has given us the cumulative distribution function (CDF) for the random variable X, which represents the waiting time to get a table at a restaurant. To find P(X ≥ ½), we need to use the CDF to calculate the probability that X is greater than or equal to ½ hours.

Using the property of the CDF for continuous distributions, we know that P(X > x) = 1 − P(X < x). So, P(X ≥ ½) = 1 − P(X < ½). Because the CDF is given as F(x) = x³/8 for 0 ≤ x ≤ 2, we first calculate F(½) = (½)³ / 8 = 1/64. Therefore, P(X < ½) = 1/64 or 0.015625.

To find P(X ≥ ½), we subtract this value from 1: P(X ≥ ½) = 1 - 1/64 = 63/64 = 0.984375. This means that the correct answer is c) 0.9844 (rounded to four decimal places).