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].
Mean is just X̄ = 12 (X1 + X2 ). The density function is evaluated by integrating over
the region where X̄ takes a particular value, and this is the area of the diagonal stripe
across the square
X2
x x+dx
(
fX̄ (x) =
X1
4x
0 ≤ x ≤ 12
4(1 − x) 12 ≤ x ≤ 1
2. Find the sample mean and standard deviation of
Z = {2.7, 3.4, 3.7, 4.2, 4.4, 5.2}
X̄ = 3.93, σ = 0.791
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}
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The sample mean is 20.3, so a 95% confidence interval for µX would be [µ − 2 √σ6 , µ +
√
2 √σ6 ] (where the factor of 6 comes from the six samples). So with 95% confidence,
we can say µX lies in the range [18.7, 22.0]
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}
The sample mean and variance is Ȳ = 14.82, s = 0.435. The width of the distribution
is then given by w = 2.78s = 1.21. The confidence interval is then [Ȳ − w, Ȳ + w] =
[13.6, 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?
The rate for the process is µ = 1/2, so the probability at least one clamper comes
to my car in time t is 1 − p0 (t), with p0 (t) = e−µt , so the resulting probabilities are
22.1%, 39.3% and 77.7%.
6. I break my office coffee mug once per year on average, and the breakages are distributed according to Poisson statistics. In the past seven years, I have broken seven
mugs. What is the probability I broke exactly one mug in each of those seven years?
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