1. Probabilities of the normal distribution. Suppose X is a normally distributed random variable with a mean of 120 and a variance of 625. Obtain the following probabilities and show the steps involved in the calculation. Use the table of the normal distribution to answer these questions. Show all steps required to obtain the answer otherwise a penalty applies. (a) P(X < 123) (b) P(X > 116) (c) P(111 < X < 124.5) (d) Find b such that P(X
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
1. Probabilities of the normal distribution.
Suppose X is a
of 120 and a variance of 625. Obtain the following probabilities
and show the steps involved in the calculation.
Use the table of the normal distribution to answer these questions.
Show all steps required to obtain the answer otherwise a penalty
applies.
(a) P(X < 123)
(b) P(X > 116)
(c) P(111 < X < 124.5)
(d) Find b such that P(X<b) = 0.67
2.Conditional probabilities.
(a) “Most students studying Economics have a laptop, hence a
student who has a laptop is most likely studying Economics”
what is wrong with this claim? Note that less than 10% of
students are studying Economics. Explain this using
conditional probabilities and the Bayes’ theorem.
In May 2020 about 4 out of 1000 people living in the UK had
contracted Covid19. The most common diagnosis test at the time
was correct at 90% for people with Covid19 (true positive). Instead,
the probability of false positive was 3%.
(b) Find the probability of the test being positive: P(T).
(c) What is the probability that a random person found to have a
positive result in May 2020 actually had Covid19, assuming
you know nothing about the person’s symptoms. Why does
this probability differ substantially from P(T|C)?
(d) In January 2021 the number of affected people in UK went up
to 28 out of 1000 people. Find the probability of having
contracted Covid19 for a random person with a positive test in
January 2021.
e) Consider a new test that gives more true positives (95%) but
also more false positive (7%). Find the probability of having
contracted Covid19 for a random person getting a positive
outcome on this new test in January 2021. Provide an
intuition to explain how P(C|T) calculated here differs from
the one in c).
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