Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x^2/16 0 ≤ x < 4 1 4 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.)Required:a. Calculate P(x<=1)b. Calculate P(0.5<= x<=1)c. Calculate P(x>1.5)d. What is the media checkout duration μ? e. Obtain the density function f(x)f. Calculate E(X).

a) P(x≤1) = 0.0625

b) P(0.5≤x≤1) = 0.0468

c) P(x>1.5) = 0.8593

d) Median (m) = 3.4641

e) f(x) =   0     x < 0

            x/8   0≤x≤4

              0      x≥4

Step-by-step explanation:

a) To calculate the probability of F(x) with x being less or equal 1 is:

P(x≤1) = F(1)

F(1) =

P(x≤1) = 0.0625

b) To calculate probability of F(x) with x being between 0.5 and 1:

P(0.5≤x≤1) = F(x≤1) - F(x≤0.5) = F(1) - F(0.5)

F(1) = 0.0625

F(0.5) = = 0.0156

P(0.5≤x≤1) = 0.0625 - 0.0156

P(0.5≤x≤1) = 0.0468

c) P(x>1.5) = 1 - P(x≤1.5) = 1 - F(1.5)

F(1.5) = 0.1406

P(x>1.5) = 1 - 0.1406

P(x>1.5) = 0.8593

d) Median is a point in the graph that divides it in half, so to determine the point, here called m:

m³ = 24

m = 3.4641

The median checkout duration is 3.4641 hours.

e) Density function of a cumulative distribution function (cdf) as well as in a continuous random variable is the first derivative of a function. Then, for this function it is:

f(x) = F'(x)

f(x) =  0        x<0

               0≤x≤4

        0         x≥4

f) E(X) =

         =

         =

        =

        =

E(X) = 2.6667

ok

step-by-step explanation:

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional