Description
1. (a) Suppose 1 in 1000 persons has a certain disease. A test detects the disease in 95% of diseased persons.
The test also ”detects” the disease in 1% of healthy persons. With what probability does a positive
test diagnose the disease?
(b) Machines M1, M2, M3 produce these proportions of a article: M1 : 5%, M2 : 25%, M3 : 70%. The
probability the machines produce defective articles is M1 : 5%, M2 : 4%, M3 : 2%. What is the
probability a random article was made by machine M1, given that it is defective?
(c) A machine M consists of three independent parts, M1, M2, and M3. Suppose that M1 functions
properly with probability 9
10 , M2 functions properly with probability 9
10 , M3 functions properly with
probability 8
10 , and that the machine M functions if and only if its three parts function. What is the
probability for the machine M to malfunction?
2. Three balls are selected at random from a bag containing 3 red, 3 green, and 4 blue balls. Define the random
variables R = the number of red balls drawn, and G = the number of green balls drawn. List the values of
(a) the joint probability mass function pR,G(r, g).
(b) the marginal probability mass functions pR(r) and pG(g).
(c) the joint distribution function FR,G(r, g).
(d) the marginal distribution functions FR(r) and FG(g).3. For the preceding problem, also determine
(a) The conditional probability mass functions pR|G and pG|R. Are R and G independent?
(b) E[R] and E[G]
(c) V ar(R) and V ar(G)
(d) cov(R, G)
4. (a) A trial consists of tossing two dice. The result is counted as successful if the sum of the outcomes is
12. What is the probability that the number of successes in 36 such trials is greater than one? What
is this probability if we approximate it using the Poisson random variable?
(b) Customers arrive at a counter at the rate of 20 per hour. Assume the arrivals have a Poisson distribution.
What is the probability that more than two customers arrive in a period of 5 minutes?
5. Approximately 20,000 marriages took place in Qu´ebec last year. Assuming that each person’s birthday is
equally likely to be any of the 365 days of the year, estimate the probability that for one or more of these
couples:
(a) both partners were born on April 1;
(b) both partners celebrated their birthday on the same day of the year.
(Hint: The Poisson random variable can be used
6. For the random variable X with density function
f(x) =
4x , 0 < x ≤
1
2
4 − 4x , 1
2 < x ≤ 1
0 , otherwise
(a) Determine the distribution function F(x), and draw the graphs f(x) and of F(x).
(b) Determine P(
1
3 < X ≤
1
2
).
(c) Determine E[X].
(d) Determine V ar(X), and σ(X).
7. For the random variables X, Y with joint density function
f(x, y) =
cxy2
(1 − x)(1 − y
2
), 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
0, otherwise
(a) For what value of c is this a joint density function?
(b) Using this value of c, compute the density functions of X and Y .
(c) What is the value of Cov(X, Y )?
(d) Determine P{X > Y }
8. The side measurement of a die manufactured by a company is a random number X that is uniformly
distributed between 1 and 1.25 cm. (You may assume the die is a perfect cube.)
(a) Determine the distribution function of the volume of the die.
(b) What is the probability that the volume of a randomly selected die manufactured by this company is
greater than 1.424?
9. Question 9 has been shifted to A3. Its solution is omitted.
10. A stick of length 1 is split at a randomly selected point X, i.e., X is uniformly distributed in the interval
[0, 1]. Determine the expected length of the piece that contains the point 1/3.