Question 6 in the attached file
6. A stick of length L is broken in two pieces at a point which is uniformly distributed on thestick?s length.(a) What is the p.d.f. of the length of the smaller piece?(b) What is the expectation of the ratio of the smaller length to the larger?
Math 180A - Homework 8
( Due: Friday Nov 18, 2:00 PM)
Reading: Sections 5.5-5.7 of the textbook
Show FULL JUSTIFICATION for all your answers. 1. Problem 5.18 of the textbook.
2. Problem 5.21 of the textbook.
3. Let X ? N (?, ?).
(a) For every t ? R, compute f (t) = E[etX ].
(b) Compute f (0), f 0 (0), and f 00 (0).
4. Problem 5.37 if the textbook.
5. Suppose U has a uniform distribution on [0, 1], and define X = ? ln(U ). Find the probability
density function of X.
6. A stick of length L is broken in two pieces at a point which is uniformly distributed on the
(a) What is the p.d.f. of the length of the smaller piece?
(b) What is the expectation of the ratio of the smaller length to the larger? 1 More Practice (Do not Turn in)
1. Let X ? N (?, ?), and define Y = X 2 . Compute the probability density function fY of Y .
2. Suppose F : R 7? [0, 1] is a function that is, 1) strictly increasing (and therefore, invertible),
2) limx??? F (x) = 0 and limx?? F (X) = 1, and 3) right-continuous. Suppose U has a
uniform distribution on [0, 1], and define X = F ?1 (U ), when F ?1 is the inverse of F . Find
the cumulative distribution function of X.
Note: Problem 5 is a special case of this problem. 1 Figure 1: Angel ? and the length of the chord D in the unit circle.
3. Let ? be a random variable with uniform distribution on [0, 2?], and let D be the length of
the corresponding chord in the unit circle as shown in Figure 1. Find the probability density
function of D. 2
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