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(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].

2

(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%.

2

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Parent category: Education