Probability And Statistical Methods For Engineers Worksheets With Answer Key - Mae 108, 2011 Page 2
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4(2) (10 points)
Consider a continuous random variable, X, defined on the interval [0, 1] with a cumulative distri-
bution function (CDF) given by F (x) = A sin(πx/2) when 0
x
1.
(a) What is the value of A?
We need F (upper bound) = 1 and thus A = 1.
(b) What is the probability distribution function, f (x)?
f (x) = dF/dx =
cos(πx/2), valid on the interval [0, 1].
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(c) What is the probability that X is larger than 1/2?
P (X
1/2) = 1
P (X < 1/2) = 1
F (1/2) = 1
sin(π/4) = 1
1/ 2
29.3%.
(d) If we know that X is less than 1/2, what is the probability that it is larger than 1/3?
Careful, this is not the probability that X is between 1/3 and 1/2, because we know that it is
already below 1/2. Instead, it is a conditional probability, namely
P (1/3
X
1/2)
F (1/2)
F (1/3)
sin(π/4)
sin(π/6)
1/ 2
1/2
P (X
1/3 X
1/2) =
=
=
=
P (X
1/2)
F (1/2)
sin(π/4)
1/ 2
which is also, it turns out, 29.3%.
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