The tabular version of Bayes’ theorem: You’re listening to the statistics podcast for two groups. Let’s call them group Cool and group Clever

a) The tabular version of Bayes’ theorem: You’re listening to the statistics podcast for two groups.
Let’s call them group Cool and group Clever.
i. Prior: Let the prior probability be proportional to the number of podcasts each group has created.
Kul has made 7 podcasts, Flink has made 4. What are they respective prior probabilities?
ii. In both groups, they draw lots to see who in the group will start the consignment. Kul has 4 boys
and 2 girls, while Flink has 2 boys and 4 girls. The broadcast you are listening to is initiated by a girl.
Update the probabilities of which of the groups you are listening to now.
iii. Group Cool bowls for the statistics within 5 minutes after the intro of 70% of their podcasts.
Gruppe Flink does not toast to its podcasts. What is the likelihood that they will toast within 5
minutes of the podcast are you listening now?
b) Bernoulli Process: You will estimate π, the percentage who identify as Jedi rather than Sith. To do
this, do an experiment with Jon and Laurits. Jon and Laurits are at Outland with you on May 4th.
”May the 4th Be With You ”. Jon hands out Sith drops, while Laurits hands out Jedi drops. Customers
choose for themselves which drops they will take. You count how many each of them gets
distributed. Jedi = 54, Sith = 29.
i. Use Jeffreys’ prior hyperparameters for π. You will find the observations in the table below. Find
the posterior probability distribution for π, and draw both pdf for the probability distribution.
ii. Calculate a 70% interval estimate (“credibility interval”) for π, draw CDF for the probability
distribution for π and mark the interval estimate on this Basket.
iii. Draw a confidence curve for π and mark the 70% interval estimate for π at this basket.
c) Poisson process: A student group on renewable energy has done a bachelor project where they
have, among other things, observed notices about electricity prices in the largest the news channels.
We will use their data to make inferences about the frequency to these postings.
i. The group observed 13 articles in the largest news channels during the last 5 months of 2021. Use
this observation with neutral prior hyperparameters for Poisson process to find a posterior
probability distribution for the rate parameter λ, average spread per month.
ii. What is the probability that there will be exactly 3 such postings next month?
iii. Find the probability distribution for+2, the waiting time for the next 2 occurrences, and calculate
a 90% interval estimate (“predictive interval”) for +2.
(d) Gaussian process: A bachelor project group in the spring of 2022 calls itself “Gærningarna på
Labben” (GL). They have tested different types of concrete, and we have been able to borrow the
data for this exam. We will look at the compressive strength of A = Leca 300 vs. B = Leca 300 with
more cement. We assume that XA, the compressive strength of a random sample concrete of type A,
follows the probability distribution XA ∼ φ (µA, σA), and corresponding to B at XB ∼ φ (µB, σB)
i. The first compressive strength measurements for concrete type A are: {x1 = 20.0, x2 = 21.5, x3 =
20.0, x4 = 20.2, x5 = 18.4} (N / mm2 ). Use neutral prior and find posterior distributions for µA and τA,
and predictive distributions for XA +.
ii. Draw a confidence curve for τA and mark an 80% interval estimate (“credibility interval”) for τA.
You get half the payoff if you draw the posterior instead probability distribution (pdf) for τA, and
marks the interval estimate in this the diagram.
iii. The next measurements for concrete type A are {x6 = 17.3, x7 = 14.9, x8 = 19.4}.
Use the previous posterior as a new prior, and find a new and updated posterior probability
distribution for µA. (You do not need to update τA and XA +.)
iv. The compressive strength measurements for concrete type B are: {y1 = 25.3, y2 = 19.7, y3 = 26.1,
y4 = 21.8, y5 = 21.8, y6 = 20.6} (N / mm2). Use neutral prior and find posterior distribution for µA.
v. Use the last posterior distributions for each µ, and determine the following hypothesis test with
significance α = 0.05:
H1: µB> µA
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