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        Psychology 406

        Assignment \#2


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1) Hays --- Chapter 1, \#18; Chapter 2, \#20; Chapter 3, \#18, \#20

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2) Suppose a fair coin is flipped four times and the number of
heads, X, is recorded.  Assume that each different sequence of heads
and tails has the same probability of occurring, e.g.,
P\{(H,T,H,T)\} = 1/16.

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a) Complete the theoretical probability distribution for X:

P(X = 0) =

P(X = 1) =

P(X = 2) =

P(X = 3) =

P(X = 4) =

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b) Graph both the probability mass function and the cumulative
distribution function for X.

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c) Calculate:

i) P(1 $\le$ X $\le$ 3) =

ii) P(X $\ge$ 4) =

iii) P(X $<$ 1) =

iv) P(1 $<$ X $<$ 2) =

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3) Now, actually flip a coin four times (or four coins once) for
twenty different trials, and record the number of heads in a
frequency table.

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a) Draw a histogram, frequency polygon, and a cumulative frequency
distribution for your results.

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b) Do your results conform to the theoretical probability
distribution?  Should they?  What would you expect to obtain in
twenty trials based on your answer in 2(a)?

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4) Consider a pointer calibrated as in the diagram below:

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a) Draw the probability density function and the cumulative
distribution function for X = spot where the pointer lands.

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b) Using the probability density function as a guide, find the
following probabilities:

P(1 $<$ X $<$ 2) =

P(1 $\le$ X $\le$ 2) =

P(2 $\le$ X) =

P(X $\le$ 3) =

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c) Represent algebraically each of the four probabilities above in
terms of the cumulative distribution function, F(a) = P(X $\le$ a).

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5) Suppose we draw a card from a normal deck \emph{and} toss a
6-sided die.

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a) What would be the sample space?  How many members?

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b) Assume all members of the sample space are equally likely.  Let A
be the event of obtaining a Queen; B the event of obtaining a 4 on
the die.

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How many members do the sets A and B contain separately?

What is P(A), P(B)?

How many members does A$\cap$B contain?

What is P(A$\cap$B)?  Are A and B independent?

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c) In what way is the assumption of equally-likely elements of the
sample space important?

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6) Suppose I have a ``fair'' 4-sided die, i.e., a pyramid with
either 1, 2, 3, or 4 spots on a side, where the probability of any
one side turning face down is 1/4.  The outcome associated with any
one trial will be the number of spots on the side of the die that is
facing down.

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a) If I toss the die four times and record the outcomes
sequentially, e.g., (1,3,2,2), how many different possible outcomes
could be obtained?

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How many ways could I observe a sequence that did not contain a
repeat of numbers, e.g., (1,3,2,4), (3,1,2,4), etc.

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b) If I toss the die three times instead of four, how many ways
could I observe a sequence that did not contain a repeat of numbers,
e.g., (1,4,3), (1,3,2), etc.

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Now, if I did not care about the order of the sequences containing
no repeats, how many different outcomes could I observe?  Put your
answer in terms of a binomial coefficient.

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c) Assume that the die is ``loaded'', i.e., the four outcomes of a
toss are not equally-likely.  Let X = 1, 2, 3, or 4 depending on the
outcome of a single toss, and suppose the probabilities are as
follows:

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P(X = 1) = 3/4

P(X = 2) = 1/8

P(X = 3) = 1/8

P(X = 4) = 0

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If the die is tossed four times and the four trials do not affect
one another, what is the probability of observing the following
sequences?

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i) (1,1,1,1)

ii) (1,4,3,1)

iii) (1,2,3,3)

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If the die is tossed four times and Y = number of 1's in the four
trials, complete the theoretical probability distribution for Y.
(Hint: How is Y related to a binomial random variable?)

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7) Suppose we pick a person at random and force a Republican or a
Democrat disclosure (only two acceptable labels).  If the
probability of responding ``Democrat'' is 3/4, write expressions for
the following (do not actually compute):

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a) In a random sample of 1,000 people, the probability that 200 will
say ``Democrat'' and 800 will say ``Republican''.

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b) In a random sample of 1,000 people, the probability that 50 or
less will say ``Democrat''.

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c) In a random sample of 1,000 people, the probability that 400 to
600 people will say ``Republican''.





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