MA10211. Statistics and Probability I. Example Sheet Six Hand in ...

MA10211. Statistics and Probability I.
Example Sheet Six
Hand in your solutions by noon, Wednesday week 8 or earlier if you
tutor so requests.
1. Suppose each page of a typed article contains at least one mistake with
independent probability p. Let X be the total number of pages with
mistakes on them in an n page article. State the exact distribution
for X and the average number of pages in the article that contain
mistakes, E(X). If n is large and p small, state an approximation for
the distribution of X
Suppose, on average, a typist makes mistakes on one in every 60 pages.
In a 20 page article, find the exact probability there are (i) no mistakes
(ii) at least two mistakes. Re-calculate these probabilities using the
approximation. Is the approximation any good here
2. Professor U.N.Ethical conducted the following cruel experiment: A rat
has to choose between four similar doors, only one of which is ‘correct’.
A correct choice is rewarded with food and an incorrect choice is punished
by a small electric shock. If an incorrect choice is made, the rat is
returned to the starting-point and chooses again, this continuing until
the correct response is made. The food is not moved between successive
trials. Let the random variable X be the number of trials required
until the first correct response is made, X thus taking values 1, 2, . . . .
Find the distribution and mean of X under the following different hypotheses:
(i) each door is equally likely to be chosen on each trial, and
all choices are made independently of others; (ii) the rat chooses with
equal probability between the doors that have not so far been tried, no
choice ever being repeated; (iii) ∗ the rat never chooses the same door
on two successive trials, but otherwise chooses at random with equal
probabilities. [Check: Means are (i) 4 (ii) 2.5 (iii) 3 1 4 .]
3. Let Y be the score on rolling a fair die. Write down the distribution of
Y . Calculate E(Y ), E(Y 2 ) and E(α Y ), where α is some constant.
If X is a discrete random variable with P(X = i) = 1/n for each
i ∈ {1, 2, . . . , n} then X is said to have the discrete uniform distribution
on the set {1, 2, . . . , n}, written X ∼ Unif{1, 2, . . . , n}. Calculate
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E(X), E(X 2 ) and E(α X ), simplifying your expressions in each case.
[Hint: Recall, 1 2 + 2 2 + 3 2 + · · · + n 2 = n(n + 1)(2n + 1)/6.]
4. A penny is tossed four times. Let X be the number of tails in the first
two tosses and let Y be the number of tails in the last two tosses.
Find the distribution of X and the distribution of Y . Deduce, with
reasons, the joint distribution for (X, Y ).
Let Z to be the number of tails in the last three tosses. Find the joint
distribution of (X, Z) [Hint: its easiest to draw a table of probabilities,
check they add to 1]. Find the marginal distribution of Z. Show that
X and Z are dependent RVs.
Find the conditional distribution of X given that Z = 3, that is the
probabilities P(X = i | Z = 3).
5. Suppose that discrete random variables X, Y have joint distribution
P(X = x; Y = y) = k(x + y) for x ∈ {0, 1, 2}, y ∈ {1, 2, 3}, where k is
a suitable constant. Find the value of k.
Find the marginal distributions for X and Y . Are X and Y independent
random variables
Calculate E(X) and E(Y ). Deduce E(2X + 3Y + 4) and E(X − Y ).
Calculate E(X − Y ) directly from the joint distribution using the ‘law
of the unconscious statistician’ and verify that this agrees with your
previous answer.
By a suitable summation, calculate P(X ≥ Y ).
6. Flaws in the plating of large sheets of metal occur at random with, on
the average, one in each section of area 10m 2 . What is the probability
that a sheet 5m by 8m will have no flaws At most one flaw An even
number of flaws ∗
http://www.maths.bath.ac.uk/∼ak257/probstat1.html
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