ACCRA TECHNICAL UNIVERSITY
2022/2023 FIRST SEMESTER EXAMINATIONS
DEPARTMENT OF APPLIED MATHEMATICS \& STATISTICS
BACHELOR OF TECHNOLOGY-STATISTICS
STA 403
PROBABILITY THEORY \& DISTRIBUTIONS
TIME: 3 HRS
ANSWER.ALL QUESTIONS
QUESTION 1
a. A probability space is given as ( \( \Omega, \mathcal{F}, \mathbb{P}) \). Explain what \( \Omega, \mathcal{F} \) and \( \mathbb{P} \) stands for.
3 Marks
Given a probability space and a collection of subsets on this space define the following
i. Algebra
2 Marks
ii. Sigma algebra
2 Marks
(ifi) Borel sigma algebra
2 Marks
b.
i. Given a probability space and \( (\Omega, \mathcal{F}, \mathbb{P}) \) and a measurable space \( (\mathbb{R}, \mathcal{B}, w) \) where \( \mathbb{R} \) is the real line, \( \mathcal{B} \) is a Borel set and \( w \) is a measure on the Borel set, define a random variable on these two spaces and write out the sigma algebra generated by the random variable.
4 Marks
ii. A die is tossed once. Write out the sample space \( \Omega \). and indicate all the subsets of the \( \Omega \) 6 Marks
a. Explain the following terms
i. Probability density function
2 Marks
ii. Probability mass function
2 Marks
iii. Cumulative distribution function
2 Marks
QUESTION2
a. Given a probability space \( (\Omega, \mathcal{F}, \mathbb{P}) \) explain the statement that two events A and B are independent.
5 Marks
b. Given that \( \mathbb{P}(A / B)=\frac{P(A \cap B}{\mathbb{P}(A)} \), show that if the two events \( A \) and \( B \) are independent then
\[
\mathbb{P}(A / B)=\mathbb{P}(A)
\]
5 Marks
c. Two factories, Factory \( A \) and Factory \( B \) manufacture goggles. 20 per cent of the goggles from factory A and 5 per cent from factory B are defective. Factory A produces twice as many goggles as Factory \( B \) every week. What is the probability that a goggle, randomly chosen from a week's production, is satisfactory?
d. Show that an event is independent of itself iff \( A=0 \). 5 Marks
e. Given \( B \in \mathcal{F} \) and \( \mathbb{P}(B)>0 \), show that the condite or \( A=1 \quad 5 \) marks probability measure on \( (\Omega, \mathcal{F}, \mathbb{P}) \)
