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        Psychology 406

        Assignment \#4


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


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

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\begin{tabular}{cc}

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

\end{tabular}

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Suppose I make three independent observations on X, i.e.,
$\mathrm{X}_{1}$, $\mathrm{X}_{2}$, and  $\mathrm{X}_{3}$.

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

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b) let Y = $(1/3)(\sum_{i=1}^{3}\mathrm{X}_{i})$.  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))

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c) E(X) = \rule{.5in}{.01in}? Var(X) = \rule{.5in}{.01in}?

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

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

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f) Suppose I wish to estimate Var(X) on the basis of
$\mathrm{X}_{1}$, $\mathrm{X}_{2}$, and  $\mathrm{X}_{3}$, and do
not have the exact probabilities associated with X.  Assume
$\mathrm{X}_{1}$ = 1, $\mathrm{X}_{2}$ = 0, and  $\mathrm{X}_{3}$ =
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?

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

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a) P(1 $\le$ X $\le$ 3) =  \rule{.5in}{.01in}?

 P(X $\ge$ 4) =  \rule{.5in}{.01in}?

 P(X $\le$ -1) =  \rule{.5in}{.01in}?

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

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

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E(Y) =  \rule{.5in}{.01in}?

Var(Y) =  \rule{.5in}{.01in}?

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What is the distribution of Y?

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Suppose I have 10 independent observations on X, i.e.,
$\mathrm{X}_{1}$, $\ldots$, $\mathrm{X}_{10}$.

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Let M = (1/10)$\sum_{i=1}^{10}$$\mathrm{X}_{i}$.

What is E(M)? Var(M)?

What is the distribution of M?

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4) Assume X is binomial with N = 100, p = 1/4.

Using the normal approximation, calculate the following quantities:

P(20 $\le$ X $\le$ 40) =  \rule{.5in}{.01in}?

P(X $\le$ 80) =  \rule{.5in}{.01in}?

P(X $\ge$ 20) =  \rule{.5in}{.01in}?

















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