Download Solutions to Quiz # 4 (STA 4032)

Solutions to Quiz # 4 (STA 4032)
1. Let X and Y have a joint probability density function
c(x + y), if 0 < x < 2, x < y < x + 1,
fXY (x, y) =
0,
otherwise
(a) Determine the value of c.
(b) Find the marginal pdf of X.
(c) Find fY |x (y) and E(Y |x).
(d) Compute P (1 < Y < 3/2|X = 1).
Solution.
(a) To determine the value of c, we need to use the property that
R∞ R∞
f
(x, y)dydx = 1. We have
−∞ −∞ XY
Z
∞
Z
∞
2
Z
Z
x+1
(x + y)dydx = c
fXY (x, y)dydx = c
−∞
Z
−∞
(xy + y 2 /2)|x+1
dx
x
0
x
0
2
2
Z
= c
(2x + 1/2)dx = 5c = 1.
0
So, c = 1/5.
(b) By definition, we have
∞
x+1
1
fXY (x, y)dy =
(x + y)dy
5
x
−∞
y 2 x+1 4x + 1
1
xy +
|
=
, 0 < x < 2.
=
5
2 x
10
Z
Z
fX (x) =
Otherwise, fX (x) = 0.
(c) By definition, we have
fY |x (y) =
fXY (x, y)
(x + y)/5
2(x + y)
=
=
, 0 < x < 2, x < y < x + 1.
fX (x)
(4x + 1)/10
4x + 1
and
Z
∞
E(Y |x) =
x+1
Z
y·
yfY |x (y)dy =
−∞
x
2(x + y)
12x2 + 9x + 2
dy =
.
4x + 1
12x + 3
(d)
Z
P (1 < Y < 3/2|X = 1) =
3/2
Z
fY |1 (y)dy =
1
1
3/2
2(1 + y)
dy = 9/20.
5
2. Assume that the weights of individuals are independent and normally distributed
with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 20
people squeeze into an elevator that is designed to hold 3400 pounds.
(a) What is the probability that the load (total weight) exceeds the design limit?
(b) What design limit is exceeded by 20 occupants with probability 0.001?
Solution. (a) Let X be an individual’s weight. Then, X ∼ N (160, 900). Let
Y = X1 + X2 + · · · + X20 , where Xi denotes the weight of the ith person. Then, Y
has a normal distribution with E(Y ) = 20E(X) = 20 · (160) = 3200 and V (Y ) =
20 · V (X) = 20 · (900) = 18000. Hence, the desired probability is
3400 − 3200
Y − 3200
> √
) = P (Z > 1.49)
P (Y > 3400) = P ( √
18000
18000
= 1 − P (Z ≤ 1.49) = 1 − 0.9319 = 0.0681.
(b) Let the design limit be x pounds. Then, we have P (Y > x) = 0.001. This is
equivalent to
x − 3200
P (Z > √
) = 0.001.
18000
√
From the Table, we find P (Z > 3.09) = 0.001. So (x − 3200)/ 18000 = 3.09 which
gives the solution x = 3614.567 ≈ 3615 pounds.