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ST903:  Statistical Methods

Assignment 1

(iii)  Suppose the distribution of lengths can be assumed to be is  Gaussian.   How large a

standard deviation is acceptable if the probability of observing a plank above 330 cm may    not be larger than 0.0001?                                                                                                    [2]

(iv) Would either of the previous two answers be changed if a plank below 270 cm also needed

to be avoided?                                                                                                                       [2]

(v) What do the previous answers tell us about the usefulness of distributional assumptions    in statistics?                                                                                                                           [2]

[TOTAL: 10]

2:    (i) Let X and Y be random variables with finite means. Show that

minE[(Y − g(X))2] = E[(Y − E[Y |X])2]

g(x)

where g(x) ranges over all functions.  E[Y |X] is sometimes called the regression of Y on

X, the “best” predictor for Y conditional on X .                                                                  [3]

(ii) Let X1  and X2  be independent standard Normal random variables, i.e. Xi  ∼ N(0, 1),i =

1, 2. Find the distribution of the random variable  .                                                [3]

[TOTAL: 6]

3: Let X and Y be independent Poisson distributed random variables with parameters θ and λ ,

respectively i.e. X ∼ Poi(θ) and Y ∼ Poi(λ).

(i) Write down the joint density function of (X,Y), indicating the possible values X and Y    can take.                                                                                                                                [1]

(U,V), indicating clearly which values U and V can take.                                                   [4]

(iii) Derive the marginal density function of U .                                                                           [4]

(iv) Interpret the result found in (iii) and state your finding as a theorem.                               [2]

(v) Derive the conditional density function of V |U  and state which known distribution it

corresponds to.                                                                                                                      [3]

[TOTAL: 14]