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[answered] Math 180A - Homework 8 ( Due: Friday Nov 18, 2:00 PM) Readi

**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

stick?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? 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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