STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 1 Tutorial Solutions
(a) Ω = {HH,HT,TH,TT}.
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(b) Ω={sss,ssc,scs,css,scc,csc,ccs,ccc}.
(c) Ω={(i,j):1≤i≤6,1≤j ≤6}. (d) Ω={t, t≥0}.
There are 24 = 16 events associated with the experiment.
F = {φ,Ω,{HH},{HT},{TH},{TT},{HH,TT},{HH,HT},{HH,TH},{HT,TH}, {HT,TT},{TH,TT},{HH,HT,TH},{HH,HT,TT},{HH,TH,TT},{HT,TH,TT}}.
3. (a) In Q1(a) the probability that two heads are thrown is 1/2 × 1/2 = 1/4 as the throws are independent.
(b) In Q1(c), assuming the dice are fair, the probability that the sum of the upper face numbers is 9 is 4/36 = 1/9. The most likely sum of the two dice is 7 (with prob 1/6).
Let X be a random variable with distribution function given by
(a) P(−21
F (x) = (x + 2)/4,
−1 ≤ x < 1 x ≥ 1.
(a) P(T ≤6)=1−e−2 =0.86466...
(b) P(T < 4) = 1 − 2/3e−4/3 − 1/3 e−1 = 0.70164...
(c) P(T =3)=F(3)−F(3−)=1/3−1/3e−1 =0.210706...
(d) P(4
provided |t| < 1.
∞r −(α+1) xαx dx
= α , providedr<α. α−r
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