Psychology 406 Assignment #4

1) Hays — Chapter 5, #24, #26, #38; Chapter 6, #2, #8, #12

2) Let X be a random variable with the following values and probabilities:

X Probability
-1  1/4
0  1/4
+1  1/2

Suppose I make three independent observations on X, i.e., X_{1}, X_{2}, and X_{3}.

a) What is the sample space for the three observations? How many members? List each possible outcome and its probability of occurrence.

b) let Y = (1/3)(∑_{i=1}^{3} X_{i}). What is the probability distribution of Y? (i.e., the sampling distribution for the mean of the three observations) (Hint: Use your listing in (a))

c) E(X) = ? Var(X) = ?

d) What is E(Y)? What is its relation to E(X)? Do you have a term for this relationship?

e) What is Var(Y)? What is its relation to Var(X)?

f) Suppose I wish to estimate Var(X) on the basis of X_{1}, X_{2}, and X_{3}, and do not have the exact probabilities associated with X. Assume X_{1} = 1, X_{2} = 0, and X_{3} = 1. What is the unbiased estimator for Var(X)? Using this unbiased estimate of Var(X), what is the estimate for the standard deviation of Y?

3) Suppose X is Normal with mean 2 and variance 9 (denoted by X ∼ N(2,9)).

a) P(1 ≤ X ≤ 3) = ? P(X ≥ 4) = ? P(X ≤ -1) = ?

b) Find the value b such that P(2 - b ≤ X ≤ 2 + b) is as close to .9 as possible.

c) Suppose I have two independent observations on X, i.e., X_{1} and X_{2}. Let Y = 4X_{1} - X_{2}.

E(Y) = ? Var(Y) = ? What is the distribution of Y? Suppose I have 10 independent observations on X, i.e., X_{1}, …, X_{10}.

Let M = (1/10)∑_{i=1}^{10}X_{i}. What is E(M)? Var(M)? What is the distribution of M?

4) Assume X is binomial with N = 100, p = 1/4. Using the normal approximation, calculate the following quantities: P(20 ≤ X ≤ 40) = ? P(X ≤ 80) = ? P(X ≥ 20) = ?