Give the probability function for the number of defective

Question

Question 1: The probability that any ball point pen is defective is 0.05 independent of other pens. The pens are packaged in boxes of 20. If a package contains x defectives, the probability that any customer will demand his/her money back is x/30.

a) Give the probability function for the number of defective pens in a package.
b) Determine the probability that a customer will demand his/her money back.
c) Given that a customer has demanded his/her money back, derive the probability function for the number of defective pens in the package.
d) Given that the customer has demanded his/her money back, determine the expected number of defective pens in the package.

Question 2:

Research has indicated that for large work places, wildcat strikes occur in the factory at random at an average rate of 0.2 per year.

a) Write down the assumptions under which the Poisson distribution would describe the proba­bility of occurrence of x wildcat strikes in a work place in a year. Briefly illustrate what these assumptions would mean in this context of wildcat strikes.

b) Assuming the Poisson model applies, what is the probability that a randomly chosen large work place will experience more than one wildcat strike in a given year?

c) A survey of 2000 large work places is undertaken. Give an expression for the exact probability that at most 30 of these work places will have experienced more than one wildcat strike in the previous year. Evaluate your answer using a suitable approximation and give a brief justification for the approximation you used.

d) If X is a Poisson random variable with parameter λ and R = 1/(X + 1), determine E(R).

Question 3:

From traffic studies, we have determined that that Y, the number of passengers (humans, not pets) in can entering a conservation area has a probability distribution given in the table below.

y (Number of passengers)	0	1	2	3	4
1(11)	2/10	4/10	2/10	1/10	1/10

The conservation area charges $4.00 per car and driver, and another $1.50 per passenger.

a) Show that E(Y) = 1.5 and Var(Y) = 1.45.

b) Let Y₁, Y₂, Y₁₀₀ be 100 independent random variables with the same distribution as Y representing data on 100 cars entering the area. If D is the total revenue (in dollars) from the 100 cars, find an approximate value for P(600 ≤ D ≤ 675)

Question 4:

a) Create a joint distribution function f_x,y(x,y) for random variables X and property that Cov(X, Y) = 0, but where X and Y are not independent. distribution satisfies these conditions.

b) Let X and Y be random variables with joint probability function fx,y (x,y) Y which has the Verify that your given by:

                 x

         0      1      2

    0  1/8   3/8   1/8

y  1   1/8  1/8    1/8

i) Find the marginal probability function for X.

ii) Find P(X -Y ≥ 1).

iii) Find Var(X - 2Y + 7).

iv) Find f(x|Y = 0), the conditional distribution of X given Y = 0.

Question 5:

U and V are independent random variables with probability functions

fv(u)= (1/2)u+1, for u = 0,1,2,3,...

fv(v) = 2(1/3)v+1, for v = 0,1, 2,3,...

a) Find M_v(t), the moment generating function for U.

b) Find M_v(t), the moment generating function for V.

c) Let, S = U + V. Find a simplified form for the probability function for S (e.g. evaluate all sums), and use this to find P(S = 5).

d) Find the moment generating function for S = U + V, and use this to find E(S).

Question 6:

Suppose X is a continuous random variable with probability density function given by:

fx(x) = α/x^α+1 for x ≥ 1,

where α > 0, is a parameter (constant) for the distribution.

a)   Find P(2 < X < 3).

b)   Find P(X > 3|X > 2). Does this distribution have a memoryless property? Explain.

c)   Given an observation, u, on a random variable U from a Uniform (0,1) distribution, explain how you would simulate an observation, x, on a random variable X from the above distribution.

Question 7:

a) The diameters of mass produced washers are assumed to be normally distributed. They are assessed by measuring with two sets of calipers. Suppose that 90% of the washers will pass through calipers with spread 1.03 cm, and 20% of washers will pass through calipers with spread 0.99 cm.

Determine the percentage of the washers that have a diameter less than 1.00 CM.

b) A measurement process can be modelled by Y = μ + E where E ∼ N(0,0.04).

In this model, Y represents the observed measurement, p represents the true (but unknown) measurement, and E represents the error in measurement. How many independent mea­surements should be made on a unit so that we can be 95% sure that the average of the measurements will be within 0.10 of the true value?

Question 8:

Capture re-capture experiments are often used to estimate population size and other characteristics of wildlife populations. In these experiments, animals are captured, tagged or marked in some way, and released. Later additional animals are captured. Some of these will be tagged and some will not.

Consider an experiment where we are interested in studying some aspects of chipmunks. Suppose there is a population of N chipmunks living in the wild in an area A single trap (harmless) is placed and chipmunks are caught one at a time, marked (if not already marked) and then released.

a) Let S_r be the number of trappings before we have captured r distinct chipmunks. Assum¬ing each time a chipmunk is trapped, each of the N chipmunks is equally likely to be the one trapped, find (with justification) E(S_r), the expected number of trappings until we have trapped r distinct chipmunks.

b) Suppose it took 19 trappings to trap 15 distinct chipmunks. Provide a reasonable estimate (with justification) for the value of N.

c) For fixed r, determine lim_N→∞ E(S_r)