Stat61 - Homework #1 (9/8/07; due 9/17/07)

 

About cumulative distribution functions (cdfs)

 

A random variable X (whatever that is) has a cumulative distribution function, denoted F:R->R (or FX if we need to emphasize which random variable it applies to). The cdf can be interpreted in these ways:

 

F(x) = fraction of values of X that are ≤ x; or, equivalently,

 

F(x) = probability that a given value of X is ≤ x.

 

1. Suppose that FX(x) = 1 e-x/10. What fraction of the values of X are

 

a. Less than or equal to 10

b. Exactly equal to 10

c. In the closed interval [ 5, 10 ]

d. In the set [1, 2] [3, 4]

e. Odd, when truncated to the next lower integer. (That is, if we choose

random values of X and drop the fractions, what fraction are odd?)

 

2. Whatever a random variable is, it ought to be possible to make random draws from its distribution. That is, we ought to be able to pick a sequence of numbers a1, a2, a3, such that in the long run, the ais are distributed in the way described by FX(x). This exercise suggests one way of doing this.

 

Suppose we make random draws this way:

(1) Pick a random number t uniformly in [0,1], perhaps by using the Excel RAND( ) function

(2) Let a = F-1 (t). (That is, find a such that F(a) = t.)

 

Assume that F(x) is still given by F(x) = 1 e-x/10.

 

a. If we choose random numbers uniformly in [0,1], what fraction of them

are ≤ 0.615487 ?

b. What is F-1(t) ?

c. If our first selection for t is 0.615487, what is our first selection for a ?

d. If we choose numbers a in this way, what fraction of them are

less than or equal to 9.557777 ?

e. In general, given x, what fraction of our random draws are ≤ x ?

 

Back to reality: Problems related to Sections 2.2 through 2.5

 

3. (based on text problem 2.2.4) Suppose that two cards are dealt from a standard 52-card poker deck. Let A be the event that the sum of the two cards is 8 (assume that aces have a numerical value of 1).

 

a. Describe a reasonable sample space S.

 

b. How many elements are in S? (Not 52.)

 

c. How many elements of S are in A?

 

d. Given reasonable assumptions, what is P(A) ?

 

 

4. (based on 2.2.32) Let A and B be any two subsets of S. Use Venn diagrams to show the following:

 

a. (A B)C = AC BC

 

b. (A B)C = AC BC.

 

(These are called DeMorgans laws.)

 

 

5. Suppose that S = { 1, 2, 3, } and that P({s}) = (2/3s) for each value of s.

 

a. Verify that P(S) = 1, by adding P({s}) for all values of s.

 

b. If A is the event that s is odd. That is, A = {1, 3, 5, 7, }. What is P(A) ?

 

 

Also, do these problems from the text.

 

Section 2.3, problems 2.3.10 and 2.3.11 (both use theorem 2.3.6);

 

Section 2.4, problems 2.4.5, 2.4.28, and 2.4.42.

(2.4.5 is a famous trick question. To clarify problem 2.4.42:

If oil is low, light flashes 99% of time.

If oil is not low, light flashes 2% of time.

Before we looked at the light, we thought that P(oil low) = 0.10.

The light is flashing. Now what is the probability that oil is low? )

 

Section 2.5, problems 2.5.1, 2.5.2, 2.5.4.

 

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