STAT 381 - 13605 Quiz 6 2006-10-25
Solution
1. [6pt] Let X have the probability density function
⎪
⎩
⎪
⎨
⎧<≤
=otherwise
xx
xf ,0
10,3
)(
2
a) Find the mean of the random variable X.
() ()
(
)
4
3
4
3
3
1
0
4
1
0
2=
⎥
⎦
⎤
⎢
⎣
⎡
=== ∫∫ ∞+
∞− xdxxxdxxxfXE
b) Obtain the cumulative distribution function.
If x<0, then F(x) = P(X≤x) =0
If x≥1, then F(x) = P(X≤x) = 1
If 0≤x<1, then
() ( ) ( )
[
]
3
0
3
0
2
3xwdwwdwwfxXPxF x
xx ====≤= ∫∫ ∞−
Hence the cumulative function of X:
()
⎪
⎩
⎪
⎨
⎧
≥
<≤
<
=
1,1
10,
0,0
3
x
xx
x
xF
c) Compute P(0.5 < X < 1.2)
( ) () () ()
875.0125.015.015.02.12.15.0 3=−=−=−=<< FFXP
2. [4pt] Let Y be N(10, 4). Determine the probabilities:
a) P( 11 < Y < 14)
Y~ N (10, 4), then μ=10, σ = 2 (or σ2 = 4 )
()
()()()
2857.06915.09772.05.0225.0
2
1014
2
10111411
1411
=−=−=<<=
⎟
⎠
⎞
⎜
⎝
⎛−
<<
−
=
⎟
⎠
⎞
⎜
⎝
⎛−
<<
−
=<<
φφ
σ
μ
σ
μ
ZP
ZPZPYP
b) P ( | Y - 10| ≤2)
(
)
()
(
)
()()() ()
6826.08413.018413.01111
2
1012
2
108128
128210210210
=−−=−−=≤≤−=
⎟
⎠
⎞
⎜
⎝
⎛−
≤≤
−
=
⎟
⎠
⎞
⎜
⎝
⎛−
≤≤
−
=
≤≤=+≤≤−=≤−
φφ
σ
μ
σ
μ
ZP
ZPZP
YPYPYP