Question

Some of the common medications to help people with insomnia fall asleep include Ambien, Lunesta, and Sonata. Ambien CR is an extended-release variation and is formulated so that an individual will fall asleep within 30 minutes. The probability density function for X, the time (in minutes) it takes to fall asleep after taking an Ambien CR tablet, is given below.

 0.05,

f(x) = -0.0025(x-30),

 0,

(a) Verify that this is a valid probability density function.

if 0 ≤ x ≤ 10, if 10 < x ≤ 30, otherwise.

(b) If a randomly selected person takes an Ambien CR tablet at bedtime, what is the probability that he will fall asleep within 5 minutes?

(c) What is the probability that the person will fall asleep between 20 and 30 minutes after taking a tablet?

(d) If it takes less than 15 minutes to fall asleep after taking a tablet, people consider the medication a success. Suppose 20 people are selected at random. What is the probability that exactly 14 people fall asleep successfully?

1. Some of the medicatie te bep pple with innifall e ine A . Lanta and Sonata Ambien CR an extenderse varit is formed that ineival will fall asleep within 30 minutes. The probability density function for the time in minutes) it to to fall asleep after taking an Ambien CR tablet is given below 00 iOS 10 f(x) = -0.00258-30). (a) Verify that this is a valid probability density functie (b) If a randomly selected person takes an Ambien CR tablet at bedtime, what is the probability that he will fall ep within 5 minutes! (c) What is the probability that the person will fall asleep between 20 3 0 me ter taking a ta (d) If it takes than 15 minutes to fl atter taking a tapee t he mediatica Suppo 20 people w edat de What is the probability that exactly 1 people fallsleep fully! 2. L. X be a cotis random variable with pdf S1 10. otherwise where the parameter is constant (with respect to x). (a) Find the constant (1) Compute the imative distribution function F(x) of X (c) U F (x) (from b) to determine PX 1/2) (d) Find E(X) and V(X). 3. Let Z be a contin random variable with Z N (0.1). (a) Find the value of P(Z -0.47). (1) Find the value of P2 2.00). Note denotes the absolute vale function (c) Findsach that P(Z >) - 09382 (d) Find the 27th percentile (e) Find the value of the critical value A) X (3,00225), PIX S325) (W) X (529). PEX 50) (c) X (3.7. 1.55). P(3.05 X 310,0). (d) 30. SEX 5. Jocks in the United States and England work very hard to keep their weight down. Many participate in weight caully t o the dirt, and regularly. The weight o male

Answer 1

30 0025 (30 -x) dn 0.0S dx + 10 30 O.5+ 30 x0.002S x20- 0.15 1.5- 1 = 1. So t is a valid pd PE x <5 05 d 0.25 PL 20x 430 30 0.0025(30-x) dx . 20 30 0.0025 2 2 2. B0-26) 0025x20 x -7 20 0-0075- 0.125 0625 O75- xS157 1- PD X7 IS PT 30 2 1- 00025 (30-x) dx 1S -1- 0.002Sx 30 x (30- 1S) + 0.0025 x - 1-1125 +0.84375 0718 723 15 2 2- # R besple fll oaslab Auccursrflly Bin (20, p)

PL 7-14) 20 (1-p) 20-14 /4 0.128 3 6 O.1834.