Stats 125 Open Book Test August 2021
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Stats 125 Open Book Test
August 2021
1. Suppose P(A) = 0.5 and P(B) = 0.35.
(a) In the case when B ∈ A find P(A u B).
(b) In the case when A and B are mutually exclusive, find P(A u B).
(c) In the case when A and B are independent find P(A u B).
2. Flip 3 coins and record the outcome of each flip. Let the random variable X be the number of Heads that occur after at least one flip has come up Tails. (Note: This means that the outcomes HHH and TTT are both outcomes in the event X = 0.)
(a) Give the probability mass function of X , f女 (x).
(b) Find the expected Value of X .
3. The American county of San Giedo is trying to fight a global pandemic by encouraging its population to take the newly developed vaccine against the virus causing the pandemic. The city of San Giedo has a total population of 2.8 million people, out of which 2 million had been fully vaccinated at the beginning of July this year. Assume that the number of vaccinated individuals remains constant through July. In Table 3 you find number of cases testing positive for the virus, being hospitalized for the virus and dying from the virus, split up by vaccination status.
|
|
Fully Vaccinated |
Not Fully Vaccinated |
|
Total Cases |
1805 |
21610 |
|
Hospitalizations |
8 |
450 |
|
Deaths |
3 |
16 |
Table 1: Vaccination, Hospitalization and Mortality data from San Giedo county.
Define the following events related to a randomly selected person who was alive in San Giedo at the beginning of July, Let
● V be the event that they have been fully vaccinated
● T be the event that they have tested positive for the virus
● H the event that they have been hospitalized for the virus
● D be the event that they have died from the virus.
(a) Use the information in the text and table above to write down the following probabilities:
P(V), P(V n T), P(V n D), P(Vc n T).
(b) Find the probabilities P(T lV), and P(T lVc ), and state their interpretation in words.
(c) You see an online post claiming that being vaccinated against the virus will not impact your probability of dying from the virus. Examine this claim based on the data above. (hint: examine whether the events dying from the virus and being fully vaccinated are independent)
(d) If you find out that a particular person in San Giedo died of the virus in July, what is the probability that that person was fully vaccinated against the virus?
4. Some people claim that it is possible to be aware that one is asleep and dreaming while it is occurring, and thereby achieving some measure of control over the content of one’s dreams. Furthermore, some people claim that this phenomenon, called Lucid Dreaming, can be deliberately provoked by a technique called MILD, which involves just intending to experience lucid dreaming once one falls asleep.
Stats 125 students Moemoea and Meng have both heard about the MILD-technique, and read a claim that it works in 8% of the cases when an inexperienced lucid dreamer tries it. They both want to examine this claim.
In all of the following assume that the probability of entering a state of Lucid Dreaming is 0.08 and stays constant throughout the experiments.
(a) Moemoea just wants to see if the phenomenon is real and decides to try the technique
until it works. Let X be the number of nights that Moemoea fails to enter a state of Lucid dreaming before succeeding.
i. Find the distribution of X with parameters.
ii. Find the expected value and variance of X .
iii. Find the probability that Moemoea completes the experiment in one week. that is; find the probability that Moemoea’s first experience of Lucid Dreaming happens on the 7th night.
(b) Meng happens to get into a state of Lucid Dreaming on the first night and she decides to
keep the experiments going until she has had a total of 4 experiences of Lucid Dreaming. Let Y be the number of nights during the experiment that Meng does not experience lucid dreaming.
i. Find the distribution of Y with parameters.
ii. Find the Expected Value and Variance of Y .
iii. Find the probability that Meng finishes the experiment in exactly three weeks. That is; find the probability that Meng’s 4th experience of Lucid dreaming happens on the 28th day of the experiment.
5. Let X be a random variable, a and b constants, and g(x) and h(x) functions of x. Show that E[ag(X) + bh(X)] = aE[g(X)] + bE[h(X)].
2022-08-15