plz help me with these 4 25. (2 points) Let X be a random variable with a Beta d

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plz help me with these 4

25. (2 points) Let X be a random variable with a Beta distribution with parameters (2.5, 1.8). Use a suitable simulation to approximate the expected valiue of log(X) with an error of less than 10 and explain why you think you have achieved this accuracy. A point will be subtracted if you use more than 100 times as many simulations as is necessary 26. (2 points) For a random variable X the harmonic mean is defined Let X be a random variable with a Beta distribution with parameters (0.8, 1.5). It is known that the harmonic mean H(X) does not exist Use a suitable simulation to demonstrate this A possible way to show this is to simulate many sample means for several different sample sizes n and show that these sample means do not ttle on a reasonable (e.g. non-zero and finite) value as n becomes large. This can be done with the summary command or with side by side borplots 27. (2 points) Suppose X and Y are independent random variables that both have uniform U(0,1) distributions. Consider the events 3 Then P(A)Use R and simulations to estimate P(B), P(AIB). P(BIA) R will automatically generate independent random variables in a simulation 28. (2 points) A breathalyzer that is used by the police to detect drunk drivers correctly identifies a drunk driver with probability 0.98 and falsely identifies a sober driver as drunk with probability 0.01. Answer the following questions: a) What is the sensitivity of this breathalyzer? What is its specificity? Look up the terms "specificity" and "sensitivity b) About one in 500 drivers are drunk. A car is stopped by the police and the driver fails the breathalyzer test. What is the probability that she is drunk?

Explanation / Answer

28:

Let D shows the event that driver is drunk and P shows the event the test gives positive results.

Sensitivity: The probability of a person testing positive, given that (s)he has the disease.

Specificity: The probability of a person testing negative given that (s)he does not have the disease.

a)

Here we have

P(P|D) = 0.98, P(P|D') = 0.01

By the complement rule,

P(N|D') = 1 - P(P|D') =1 - 0.01 =0.99

P(N|D) = 1 - 0.98 = 0.02

The sensitivity of the test is: 0.98

The specificity of the test is: 0.99

(b)

Since one in 500 drivers are drunk so

P(D) = 1/500 = 0.002

By the complement rule,

P(D') = 1 - P(D) = 1 -0.002 = 0.998

We need to find the probability P(D|N). By the Baye's theorem the required probability is

P(D | N) = [ P(N|D)P(D) ] / [ P(N|D)P(D) +P(N|D')P(D') ] = [0.02 * 0.002 ] / [ 0.02 * 0.002 + 0.99 * 0.998 ] = 0.00004 / [0.00004 + 0.98802 ] = 0.00004 / 0.98806 = 0.00004048

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