BU7527
Example sheet — 3
Mike Peardon — mjp@maths.tcd.ie
School of Mathematics, TCD
Wednesday, 30th September
Try to answer these questions before tomorrow’s lectures. We will go through solutions
in class.
1. Find the probability density function of X̄, the mean of two independent random
numbers X1 and X2 that are uniformly distributed in [0, 1].
2. Find the sample mean and standard deviation of
Z = {2.7, 3.4, 3.7, 4.2, 4.4, 5.2}
3. Find a 95%(2σ) confidence interval for the mean µX of a normally distributed random
number X with standard deviation σ = 2 from the sample
X = {15, 21, 20, 17, 24, 25}
4. Student’s t-distribution with four degrees of freedom says the 95% confidence interval
of a normally distributed random number with unknown variance σ has width w =
2.78s where s2 is the sample variance. Use this to find a 95% confidence interval for
the mean µY determined from the sample
Y = {13.2, 14.5, 14.8, 15.6, 16.0}
5. I park my car illegally outside College. Clampers arrive according to a Poisson process
once every two hours on average. What is the probability I will be clamped if
a. I leave my car for half an hour?
b. I leave my car for an hour?
c. I leave my car for three hours?
6. I break my office coffee mug once per year on average, and the breakages are distributed according to Poisson statistics. In the past three years, I have broken four
coffee mugs. What is the probability I broke them all in the same year?
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