Answer all.

please give explanation

Suppose the blood of 1000 persons has to be tested to see which ones

are infected by a (rare) disease. Suppose that the probability that the test10.5 Exercises 147

is positive is p = 0.001. The obvious way to proceed is to test each person,

which results in a total of 1000 tests. An alternative procedure is the following.

Distribute the blood of the 1000 persons over 25 groups of size 40, and mix

half of the blood of each of the 40 persons with that of the others in each

group. Now test the aggregated blood sample of each group: when the test is

negative no one in that group has the disease; when the test is positive, at

least one person in the group has the disease, and one will test the other half

of the blood of all 40 persons of that group separately. In total, that gives 41

tests for that group. Let Xi be the total number of tests one has to perform

for the ith group using this alternative procedure.

a. Describe the probability distribution of Xi, i.e., list the possible values it

takes on and the corresponding probabilities.

b. What is the expected number of tests for the ith group? What is the

expected total number of tests? What do you think of this alternative

procedure for blood testing?

10.10 Consider the variables X and Y from the example in Section 9.2

with joint probability density

f(x, y) = 2

75

2x2y + xy2 for 0 ? x ? 3 and 1 ? y ? 2

and marginal probability densities

fX(x) = 2

225

9x2 + 7x

for 0 ? x ? 3

fY (y) = 1

25(3y2 + 12y) for 1 ? y ? 2.

a. Compute E[X], E[Y ], and E[X + Y ].

b. Compute E

X2

, E

Y 2

, E[XY ], and E

(X + Y )2

,

c. Compute Var(X + Y ), Var(X), and Var(Y ) and check that Var(X + Y ) =

Var(X) + Var(Y ).

10.11 Recall the relation between degrees Celsius and degrees Fahrenheit

degrees Fahrenheit = 9

5 degrees Celsius + 32.

Let X and Y be the average daily temperatures in degrees Celsius in Amsterdam and Antwerp. Suppose that Cov(X, Y ) = 3 and ?(X, Y )=0.8. Let T

and S be the same temperatures in degrees Fahrenheit. Compute Cov(T,S)

and ?(T,S)

16. Let X be a geometric random variable with parameter p. Take a large integer n, define Y = X, and set A = np. Show that Y is approximately an exponential random variable with parameter A, in the sense that P( Y a)~l -e-da for all a > 0. This approximation becomes more and more exact as n - co. p - 0 with np = > fixed. (HINT: Use the formula for the sum of a finite geometric series and the relation ex = limn-+co (1 + x)".)4. (15 points) The random variable W has a gamma distribution with parameters (A, a), where A > 0, and a > 0. Its probability density function is fw (w) = W 20 r(a) T(a) = It's moment generating function is MW (0) = where f