MAS1302 – Computational Probability and Statistics
Additional Exercises 3 (Not assessed)
These questions are designed for use in and around the third Problems Class. You do not need to hand in any
solutions, but the questions are good practice for the material covered being covered at the time.
1. Suppose U1 and U2 are two independent Uniform(0,1) random variables, and let Y=U1+U2. In lectures
we considered a geometrical proof of Pr(Y<1.5) = 7/8 = 0.875 (Example 3.2.1 Question 1).
(a) Without using a geometrical argument like the one used in lectures, state Pr(Y < 1).
(b) Back up your answer to part (a) with a geometrical proof.
2. Suppose U1 and U2 are two independent Uniform(0,1) random variables, and let Y=U1+U2. Evaluate
Pr(Y<0.25) justifying your answer with a geometrical proof.
3. Suppose U1 and U2 are two independent Uniform(0,1) random variables.
(a) Without using a geometrical argument like the one used in lectures, state Pr(U1<U2).
(b) Back up your answer to part (a) with a geometrical proof.
4. Suppose U1 and U2 are two independent Uniform(0,1) random variables.
(a) Evaluate Pr(U22 < 0.5).
(b) Evaluate Pr(U22 < U1).
5. Suppose U1 and U2 are two independent Uniform(0,1) random variables.
Evaluate Pr(U12 + U22 < 1).
6. Suppose U1 , U2 and U3 are three independent Uniform(0,1) random variables.
Evaluate Pr(U12 + U22 + U32 < 1).
7. Suppose U1 and U2 are two independent Uniform(0,1) random variables, and let Y=U1+U2.
(a) Evaluate the p.d.f. of Y. (This is Example 3.2.1 Question 2).
(b) Using your answer to part (a), evaluate the c.d.f. of Y. (This is Example 3.2.1 Question 3).
8. Suppose X1 ~ Bern(0.4);
X2 ~ Bern(0.6);
X3 ~ Bin(4, 0.4);
are independent random variables. Which of the following statements are true?
(a)
(b)
(c)
(d)
X4 ~ Bin(4,0.6)
X1+X2 is a Binomial random variable;
X1+X3 is a Binomial random variable;
X3+X4 is a Binomial random variable;
X3 can be expressed as the sum of four independent Bernoulli random variables.