Download Problems on Exam 3 material

More Problems on Exam 3 Material
1.
3
(10 points) Suppose X is uniformly distributed on the interval 0 < x < 8 and Y = - X. Find the
probability density function g(y) of Y.
3
3
Solution. Let G(y) be the cumulative distribution function of Y. G(y) = Pr{Y  y} = Pr{- X  y} = Pr{- y  X} =
d
d
Pr{- y3  X} = 1 - Pr{X < - y3} = 1 – F(- y3). So g(y) = [G(y)] = [1 - F(- y3)] = - F '(- y3) (- 3y2) =3y2 f(- y3). One has
dy
dy
3
f(x) = 1/8 for 0 < x < 8 and 0 otherwise. Furthermore 0 < - y3 < 8  - 8 < y3 < 0  - 2 < y < 0. So g(y) = y2 if – 2 < y < 0
8
and 0 otherwise.
2.
(10 points) The pressure P of the gas in a container is related to the volume V by the formula P = 1/V.
Suppose the volume is not known exactly and we treat it as a random variable with density function
for 1 < v < 2
 v - 1
for 2 < v < 3
f(v) =  3 - v
. Find the expected value of the pressure P.
 0
for v < 1 or v > 3

Solution. E(P) = E(1/V) =
-
(3ln v – v)
3
|2
1
2
1
3
1
2
1
3
3
2

 v f(v) dv = 
 v (v – 1) dv + 
 v (3 - v) dv = 
 (1 – v ) dv + 
 (v - 1) dv = (v – ln v)|1 +
1
2
1
2
= (2 – ln 2) – (1 – ln 1) + (3ln 3 – 3) – (3ln 2 - 2) = 3ln 3 – 4ln 2 = ln(27/16) = ln 1.6875  0.523.
3. Tom is going fishing. The times between successive fish catches are exponentially distributed with
mean 20 minutes and these times are all independent of each other. He is going to either fish for an
hour or until he catches three fish, whichever comes first.
a. (8 points) What is the probability that Tom will catch at least three fish in the next hour?
b. (8 points) What is the probability that he catches three fish before an hour elapses?
Solution. a. Let N(t) = number of fish he has caught after he has been fishing for a time t. N(t) is Poisson with
parameter t where  is the rate he catches fish. In this case  = 3 / hr. In general, Pr{N(t) = n} = (t)ne-t/n!. In this
case we want Pr{N(1)  3} =1 – Pr{N(1) = 0, 1, or 2} = 1 – (30e-3/0! + 31e-3/1! + 32e-3/2!) = 1 - (1 + 3 +9/2)e-3 = 1 –
(17/2)e-3 = 0.5768. b. Note that he catches three fish before an hour elapses precisely if the number of fish he would
have caught in an hour is at least three. So a and b have the same answer.