Mathematical Probability (MATS2105)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2014
UNIVERSITY OF BAMENDA
FIRST SEMESTER EXAMINATION
MATS 2105 MATHEMATICAL PROBABILITY
ANSWER ALL QUESTIONS: TIME ALLOWED: 2 hours
1.
Given that A and B are independent events, show that A
l
and B
1
are independent
events. (5 points)
2.
Given that X and Y are independent random variables, show that E(XY) =
E(X)E(Y). (5 points)
3.
A team of five is chosen at random from seven girls and eight boys.
(a) in how many ways can the team be chosen if:
(i) there are no restrictions
(ii) there must be more boys than girls on the team with at least one of
the genders present
(b) if a particular boy refuses to be in the team with a particular girl, in how
many ways can the team be formed?
(10 points)
4. A continuous random variable X has cumulative distribution function F(x) defined
by
(a)
Find the probability distribution function of X, f(x) and sketch the graph of y =
f(x).
(b)
Find E(X) and var(X).
(10 oints)
5. The random variable X is said to have a Weibull distribution with parameters
and
(
>0, > 0) if the probability distribution function of X is:
(a)
Find the mean of this distribution.
(b)
Find then variance of X.
(10 points)
6. A nuts company markets cans of deluxe mixed nuts containing coconuts, cashews and
peanuts. Suppose the net mass of each can is exactly one kilogram, but the mass
contribution of each type of nut is random. Because the three masses sum to one, a
joint probability model for any two gives all necessary information about the mass of the
third type. Let X = the mass of coconuts in the can and Y = the mass of cashews. Let
the joint distribution function for (X, Y) be:
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f(x, y) =
24 0 ≤ ≤ 1,0 ≤ ≤ 1, + ≤ 1
, ℎ
(a) Show that this is indeed a probability distribution function for coconuts and
cashews.
(b)
Find the probability that the two types of nuts together make up at most 50% of
the can.
(c)
Find the marginal probability density function for coconut.
(10 points)
7. The random variable X is distributed normally with mean and variance 6, and the
random
variable Y is normally distributed with mean 8 and variance
2
.
2X-3Y is distributed
normally
with mean — 12 and variance 42.
Find,
(a)
The value of and the value of
(b)
P (X > 8)
(c)
P( Y < 9)
(d)
P
(-4 < 3X — 2Y < 7)
(10 points)
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