MATH 3680.002
Homework #10 Solutions
p. 200 7.1: Find the probability density function f for the continuous random
variable
X with the cumulative distribution function
0
for x < 0
2
x
F(x) = for 0 x < 2
4
for x 2
1
Solution:
F'(x) = 0 for x
x
F'(x) = 2 =
4
F'(x) = 0 for x
0
x
f (x) = 2
0
< 0
x
for 0 x < 2
2
2
for x < 0
for 0 x < 2
for x 2
MATH 3680.002
Homework #10 Solutions
p. 201 7.3: Find each of the following for the density function
3 (2x x 2 )
f (x) = 2 0
(a)
(b)
(c)
(d)
(e)
for 0 < x < 1
elsewhere
The cumulative distribution function F
F(0.25)
P(X0.25)
F(0.75)
P(0.25<X0.75)
Solution:
(a) for x 0
F(x) = 0
for 0 < x < 1
x
3 2
3 2
1 3
3x 2 x dx = 2 x 2 x 0
0
x
x
3
F(x) = (2x x 2 ) dx =
0 2
3 2
1
x x3
2
2
for(x 1)
F(x) = 1
0
for x 0
3
1
F(x) = x 2 x 3 for 0 < x < 1
2
2
1
for x 1
F(x) =
2
11
3 1 (b) F(0.25) = 2
4
2
4
3
=
3
1
12
1
11
=
=
32
128
128
128
128
1
1
3
3
1 11
117
(c) P(X 0.25) = (2x x 2 ) = x 2 x 3 = 1
=
2
2 0.25
128
128
0.25 2
2
3
3 3
1 3
27
27
108
27
81
(d) F(0.75) = =
=
=
2
4
2
4
32
128
128
128
128
81
11
70
35
(e) P(0.25 < X 0.75) =
=
=
128
128
128
64
MATH 3680.002
Homework #10 Solutions
p. 201 7.6: Find each of the following for the density function
ex
f (x) = 0
(a)
(b)
(c)
(d)
(e)
for 0 x < for 0 < x
The cumulative distribution function F
F(0)
F(1)
P(X1)
P(1<X2)
Solution:
(a) F(X) =
x
x
0
x
e
f (x) =
f (x) +
0
f (x) =
0
x
+
0
0
x
F(X) = [ex ] 0 + 0 = ex (e 0 ) + 0 = 1 ex
1 ex for 0 x < F(X) = for 0 < x
0
(b) F(0) = 1 e0 = 1 1 = 0
(c) F(1) = 1 e1 .632120559
(d) P(X 1) = 1 P(X 1) = 1 (1 e1 ) = e1 .367879441
2
(e) P(1 < x 2) =
e
1
x
= [ex ]
P(1 < x 2) .232544158
2
1
= e2 (e1 ) =
1
1
e 1
2 =
e
e
e2
MATH 3680.002
Homework #10 Solutions
p. 204 7.16: Find the median of the random variable given in Exercise 7.6 on page
201:
ex
f (x) = 0
Solution:
F(x) = 1 ex (see previous exercise)
Median = 1 ex = 0.50
Median = ex = 0.50
Median = ex = 0.50
Median = e x = 2
Median = ln(e x ) = ln(2)
Median = x = ln(2) .693147181
for 0 x < for 0 < x