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F71SM STATISTICAL METHODS
Tutorial on Section 4 SPECIAL DISTRIBUTIONS (discrete distributions)
1. Let X ∼ b(n,p) and Y ∼ b(n,1−p). Note that P(X ≥ k) = P(Y ≤ n−k). Hence, using NCST, find the probabilities that in a series of 20 independent, identical Bernoulli trials with P (success) = 0.6, we obtain (i) 12 or more successes; (ii) 12 or fewer successes.
[i) 0.5956, ii) 0.5841]
2. Let X ∼ P (λ). Verify the following recursion formula.
P(X =x+1) = λ P(X =x), x=0,1,2,…
Use this result to find the mode of the distribution (in this context, the mode is the value
for which P (X = x) ≥ P (X = x + 1) and P (X = x) ≥ P (X = x − 1)).
3. A plant, whose constitution is known to be either A or B, is tested by raising n offspring from its seeds. If the constitution is A then each offspring has probability 1/4 of being white and 3/4 of being red; if B then each offspring has equal probability 1/2 of being white or red.
Using a binomial model, write down an expression for the probability pA(r, n) that exactly r of n offspring of an A plant are white, and similarly pB(r,n) for the offspring of a B plant.
If three offspring are raised from an A plant and three are raised from a B plant, find the probability that the plants produce the same number of white offspring. [0.266]
4. Edward is a learner driver whose successive attempts to pass the driving test can be regarded as independent trials, each with probability 0.6 of success (he is not an effective learner…).
(a) Find the probability that Edward needs more than 3 attempts to pass the test. [0.064]
(b) Edward’s dad is a generous sort. He offers Edward a present (in units of £1000) of 0.9k, where k is the number of attempts he requires to pass the test. Find the expected value of the present. [£843.75]
(c) Find the probability that Edward needs more than 4 further attempts to pass, given that he has had 2 unsuccessful attempts already. [0.0256]
5. The records of an airline company reveal that 4% of passengers with reservations on flights from London to fail to turn up for their flight. To increase income the airline accepts 310 reservations for such a flight to be operated using an aircraft which has only 300 seats.
Using a Poisson approximation to the binomial distribution, find the (approximate) prob- ability that the airline will be unable to allocate a seat to each passenger with a reservation who turns up for a flight and so will have to ‘bump’ at least one passenger off. [0.209]
6. Let X ∼ P(λ) and let Z be the standardised version Z = √ Z satisfies
√ t/√λ ln(MZ(t)) = −t λ+λ e −1
. Show that the mgf of
and hence show that MZ(t) → exp(t2/2) as λ → ∞.
Noting that exp(t2/2) is the moment generating function of the standard normal distri- bution N(0,1), what does this result imply about the limiting distribution of Z? What does it imply about the asymptotic distribution of X ∼ P (λ)?