Respuesta :
Answer:
(a) The value of fₓ (9.5) is 0.125.
(b) The value of fₓ (10.5) is 0.50.
Step-by-step explanation:
Let X denote delivery time of the mail delivered by Alice and Y denote delivery time of the mail delivered by Bob.
It i provided that:
[tex]X\sim U(9, 11)\\Y\sim U(10, 12)[/tex]
The probability that Alice delivers the mail is, p = 1/4.
The probability that Bob delivers the mail is, q = 3/4.
The probability density function of a Uniform distribution with parameters [a, b] is:
[tex]f(x)=\left \{ {{\frac{1}{b-a};\ a, b>0} \atop {0;\ otherwise}} \right.[/tex]
The probability density function of the delivery time of Alice is:
[tex]f(X_{A})=\left \{ {{\frac{1}{b-a}=\frac{1}{2};\ [a, b]=[9, 11]} \atop {0;\ otherwise}} \right.[/tex]
The probability density function of the delivery time of Bob is:
[tex]f(X_{B})=\left \{ {{\frac{1}{b-a}=\frac{1}{2};\ [a, b]=[10, 12]} \atop {0;\ otherwise}} \right.[/tex]
(a)
Compute the value of fₓ (9.5) as follows:
For delivery time 9.5, only Alice can do the delivery because Bob delivers the mail in the time interval 10 to 12.
The value of fₓ (9.5) is:
[tex]f_{X}(9.5)=p.f(X_{A})+q.f(X_B})\\=(\frac{1}{4}\times \frac{1}{2})+(\frac{3}{4}\times0)\\=\frac{1}{8}\\=0.125[/tex]
Thus, the value of fₓ (9.5) is 0.125.
(b)
Compute the value of fₓ (10.5) as follows:
For delivery time 10.5, both Alice and Bob can do the delivery because Alice's delivery time is in the interval 9 to 11 and that of Bob's is in the time interval 10 to 12.
The value of fₓ (10.5) is:
[tex]f_{X}(10.5)=p.f(X_{A})+q.f(X_B})\\=(\frac{1}{4}\times \frac{1}{2})+(\frac{3}{4}\times\frac{1}{2})\\=\frac{1}{8}+\frac{3}{8}\\=0.50[/tex]
Thus, the value of fₓ (10.5) is 0.50.
