STA202: STATISTICS AND PROBABILITY II - FUO - 2019 SESSION - EXAM PAST QUESTIONS (FUOEXAMS.BLOGSPOT.COM)
STA202: STATISTICS AND PROBABILITY II
(FEDERAL UNIVERSITY OTUOKE, 2018/2019 SESSION 2ND
SEMESTER)
INSTRUCTION: ANSWER 4 QUESTIONS ONLY
1. QUESTION ONE
a. A warehouse
contains ten machines, five of which are defective. A company selects six of
the machines at random, thinking all of them are in working conditions.
i.
Identify the random variable X for the problem
ii.
State the probability function of the random variable X and define all parameters involved
iii.
What is the probability that:
1. All six of
the machines are non-defective?
2. At most two
of them are defective?
b. A fair coin
is tossed four times. Obtain the
i.
Probability distribution for the random variable Y representing the number of heads
appearing during the experiment
ii.
Cumulative distribution function of Y,F(Y)
2. QUESTION TWO
a. Let X have
a density function given as: f(x) = {kx(x – 1)2;
0 ≤ x ≤ 2 where k is
a constant. Find
i.
The value of k such that f(x) is a valid probability
density function.
ii.
P(0 ≤ X ≤ 
)
iii.
F(x)
b. Find the
mean and variance of a random variable, Y having the distribution
f(y) =1/(k2 –k1); k1 ≤ y ≤ k2
0; otherwise
K1
and k2 are minimum and maximum values of y respectively
3. QUESTION THREE
a. Q3
i.
Find the moment generating function of a random
variable X, having the distribution below:
f(x) = {pqx-1; x = 1,2,3, …
0; otherwise
Where
p + q = 1
ii.
Use your result in 3(a)(i) above to determine the
first and second moment of X about
the origin.
b. Consider the
probability distribution below:
|
x |
21 |
22 |
23 |
24 |
25 |
26 |
|
f(x) |
0.05 |
0.20 |
0.30 |
0.25 |
0.15 |
0.05 |
Obtain E(X2
– 5) and E(2X3 + 2).
c. Define the
following terms:
i.
Random variable
ii.
Kth moment about the mean
iii.
Mathematical Expectation of a random variable
4. QUESTION FOUR
a. Define the
following random variables
i.
Binomial
ii.
Geometric
iii.
Hyper-geometric
iv.
Poisson
v.
Normal
5. QUESTION FIVE
a. Define
Chebyshev’s inequality
b. A historic
data shows that the distribution of a random variable can be modeled by a
probability density function f(x) = 20e-20(x-12.5) ,x ≥ 12.5, calculate
the following
i.
P(x > 12.6)
ii.
P(12.5 < x < 12.6)
6. QUESTION SIX
a. State the central
limit theorem
b. State the
convolution of the pdf’s of the random
variables X1 and X2
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