STAT 3023/3923
Semester 2 Statistical Inference 2021
Week 1 Tutorial
1. Write down the sample space for each of the following senarios.
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(a) A fair coin is thrown twice.
(b) A car passes through a sequence of 3 traffic lights. At each light the car stops ‘s’ or continues ‘c’.
(c) Two dice are rolled and the upper face numbers recorded.
(d) Recording the time until an electronic component fails.
2. If a fair coin is thrown twice write down the corresponding events space F. How many different events belong to F?
3. Computing probabilities with equally likely outcomes.
(a) In Q1(a) what is the probability that two heads are thrown?
(b) In Q1(c), assuming the dice are fair, what is the probability that the sum of the upper face numbers is 9? What is the most likely sum of the two dice?
4. Let X be a random variable with distribution function given by
(a) P(−21 < X ≤ 21), (b) P(X = 0) and
(c) P(X = 1).
(x + 2)/4, 1,
−1 ≤ x < 1 x ≥ 1.
5. The following distribution function was proposed as a model for the duration (T in minutes) of phone calls,
F(t) = 1 − 23e−t/3 − 13e−[t/3], t > 0,
where [x] denotes the integer part of x. Sketch the distribution function. Calculate
that probability that the call duration will be
(a) not more than 6 minutes, (b) less than 4 minutes,
(c) equal to 3 minutes,
(d) between 4 and 7 minutes.
6. The positive random variable Y has a log-normal distribution if log(Y ) is normal N(μ,σ2). Show that the rth moment of Y about the origin is exp(rμ + 12r2σ2). Find Var(Y ).
7. Show that the random variable with probability density function f(x)= 21 e−|x|, −∞
where α > 0.
(a) Show that E(Xr) exists if r < α. (b) Find E(X) when α > 1.
8. A random variable, X, has a Pareto distribution if it has probability density function
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