Spring 2008

Equally weighted problems. Your best problem counts double.

1. Take-home problem is due tomorrow.

2. Jar A contains 3 red and 7 white marbles; Jar B contains 4 red and 2 white marbles. A jar is chosen at random. A marble is selected from the chosen jar and placed in the other jar. A marble is then selected from this second jar. What is the probability it is white?

3. Let X₁ and X₂ have independent distributions b(4,0.3) and b(7,0.3). Find the moment generating function of Y=2X₁+3X₂.

4. A population random variable X has the distribution

x        1        2        fX(x)        0.9        0.1

(a) Let (X₁, X₂, X₃) be a random sample of size 3 of X. Complete the second column of the table, showing the probability distribution of (X₁, X₂, X₃).

(x1,x2,x3)        f(x1,x2,x3)        [`x]        m2        v     s^2    (1,1,1)                                        (1,1,2)                                        (1,2,1)                                        (1,2,2)                                        (2,1,1)                                        (2,1,2)                                        (2,2,1)                                        (2,2,2)                                                1.000

(b) Complete the remaining columns above, showing [`x], m<sub>2</sub>, v, and s<sup>2</sup> for each sample point. Recall m<sub>2</sub>=[1/n]åx<sub>i</sub><sup>2</sup>, v=m<sub>2</sub>-[`x]², and s²=nv/(n-1).

(c) Complete the following table showing the distribution of [`X] on the real line.

[`x]                                                                       f[`X]([`x])                                     1.000

(d) Create a table showing the distribution of the statistics S² and V.

5. Let (X₁,X₂,...,X₆) be a random sample from the standard normal distribution N(10,25). Let

W= 6 å i=1  (Xi - 10)2.

Find P(W > 361.25).

6. A random sample is taken from N(m,s²), and the following observations are recorded.

13.1    5.1    18.0    8.7    16.5    9.8    6.8

Find a 90% confidence interval for s².

7. Let X be the number of alpha particles counted by a Geiger counter during one minute. Assume that the distribution of X is Poisson with a mean of 9658. Determine (approximately) P(9564 < X < 9712).