Part -1:
1. Calculate the following probabilities, showing your working in your answer. (You can use R to check the values, but your answer must show details of the calculation, not just 11 output.)
(a) For X∼ Bin(8, 0.2), calculate Pr(X ≤ 2).
(b) For X ∼ Geom(0.3), calculate Pr(4 ≤ X < 9).
(c) For X ∼ Po(4), calculate Pr(X ≥ 3).
2. A random variable X takes the values {1, 2, 3, 4} The probability mass function is defined by
Pr(X = x) = A + Bx2 for x = 1, 2, 3, 4
where A and B are constants.
(a) i. Show that B = 1/30(1 -4A) necessarily in order for the probability mass function to be well defined.
ii. Explain why the probability mass function is not well defined when A = 1 and B = -1/10.
(b) When A = 1/3 calculate:
i. E[X]
ii. Var(X)
3. Hospitals are assessed for cleanliness by taking swaps from the hands of staff at random. The swabs are cultured in a laboratory and tested for the presence of drug-resistant bacteria. In total, 20 swabs are taken per hospital. The hospital fails the cleanliness assessment if 2 or more swabs indicate the presence of drug-resistant bacteria. Calculate the probability that:
(a) the hospital fails the assessment if 5% of staff carry drug-resistant bacteria on their hands;
(b) the hospital does not fail even if 15% of staff carry such bacteria?
4. Suppose that the number of times during a year that an individual catches a cold can be modelled by a Poisson random variable with an expectation of 4. Further suppose that a new drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for 80% of the population, but has no effect on the remaining 20% of the population. Calculate:
(a) the probability that an individual taking the drug has 1 cold in a year if they are part of the population which benefits from the drug;
(b) the probability that an individual has 1 cold in a year if they are part of the population which does not benefit from the drug;
(c) the probability that a randomly chosen individual has 1 cold in a year if they take the drug;
(d) the conditional probability that a randomly chosen individual is in that part of the population which benefits from the drug given that they had 1 cold in a year during which they took the dnig.
5. Two random variables X and Y have joint probability mass function defined by
P X,Y (X, Y) = {0 C|X - Y| when x {-2, -1,0,1,2} and y = ∈ {-2, -1,0,1,2} otherwise
where C is a fixed constant.
(a) Determine the value of C and write down the joint pmf of X, Y in a table.
(b) Are X and Y independent? Justify your answer.
(c) Write down the conditional pmf py|x(y|X = 1).
(d) Find Var(X).
6. No help given. This question is intended to be slightly harder than the others. Suppose two numbers are drawn at random with replacement from the set (1,2, ... ,n,} and let Z be the maximum of the two numbers. Find E[Z].
7. No help given. This question is intended to be slightly harder than the others. Suppose X1,...................X4 are IID with Xi ∼ Po(A). Let Y = 1/4(X1 +.......... + X4).
Find Pr (Y < 1/2).
Part -2:
1) The differential equation:
xy' = y - Y2 + x2
can be solved by setting y = xΦ'/Φ, where Φ(x) is a new variable.
a) Using this substitution, find the differential equation for Φ(x) an expression for Φ(x).
Hint: The expression for y is both a product and a quotient, need to use both the product rule and the quotient rule.
b) From this, construct the general solution y(x) of the original that it involves only one arbitrary constant.
c) Find the solution for y(x) which satisfies y(1) = 0.
2) For each of the following equations (which are either in terms of y(x) or x(t)), find the general solution and then the particular solution which satisfies the given conditions:
a) x" = - x' + 2e-t + 1 with x(0) = x'(0) = 0.
b) 4y" - 12V+ 9y = 0 with y(0) = 2 and y1(0) = 1.
c) x" + x' - 2x = 0 with x(0) = 2, and x → 0 as t → +∞.
d) y" + 4V+ 8y = 0 with y(0) = 0 and y'(0) = 1.
3) A famous mathematical model for an epidemic, e.g. a flu outbreak, is the "SIR" model. The population of interest is divided into 3 sub-populations: those who are susceptible to the disease S(t) (but have not yet caught it), those who are currently infected 1(t), and those who have recovered R(t) (and are no longer susceptible).
The rate at which the susceptible population changes is given by:
dS/dt = -αSI ( α = constant > 0).
The rate at which the recovered population changes is given by:
dR/dt = βI (β = constant > 0).
The rate at which the infected population changes depends on both the rate of susceptible people getting infected and the rate of infected people recovering:
dI/dt = αSI -βI.
a) Construct the differential equation for I(S) (by noting that 1/S = dI/dS).
b) Find the solution for I(S), given the condition that the epidemic begins with one infected person and N susceptible people.
c) Taking N = 500, α = 0.005 and β = 0.5, determine the maximum number of people that become infected during this epidemic.