Question

The waiting time (in weeks) for an appointment to see a psychologist can be modeled by a random variable X with pdf f(x)={2500x3​0​ if 0⩽x⩽10 otherwise. ​ (a) Find the time (in days) within which 90% of patients can get an appointment.

Answer 1

To determine within which time 90% of patients can get an appointment, we integrate the given pdf to find the CDF and solve for x when CDF equals 0.9. After solving, we find that 90% of patients can get an appointment within approximately 6.76 days.

Step-by-step explanation:

To find the time within which 90% of patients can get an appointment, we need to determine the value of x for which the cumulative distribution function (CDF) of X equals 0.9. The probability density function (pdf) is given by f(x) = 2500x3 for 0 ≤ x ≤ 10. The CDF is found by integrating the pdf:

1. Calculate the indefinite integral of f(x) which will give us the CDF F(x) = ∫ f(x) dx = ∫ 2500x3 dx = 625x4 + C, where C is the integration constant.
2. Since f(x) = 0 for x < 0, we know that F(0) = 0, which tells us that C = 0. Therefore, F(x) = 625x4 for 0 ≤ x ≤ 10.
3. Solve for x when F(x) = 0.9. The equation is 625x4 = 0.9.
4. Find the fourth root of both sides to solve for x, which yields x approximately equal to 0.965.
5. Convert x from weeks to days by multiplying by 7 (since there are 7 days in a week). x in days = 0.965 * 7 ≈ 6.76 days.

Therefore, within approximately 6.76 days, 90% of patients can get an appointment with the psychologist.