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Introduction to Probability & Statistics 2019
Week 4 Problem Sheet
1*. What is the probability that exactly 3 heads are obtained in 5 tosses of a fair coin?
2*. A fair die is thrown until the sum of the results of the throws exceeds 6. The random variable X is the number of throws needed for this. Let FX be the distribution function of X. Determine FX (1), FX (2) and FX (7).
3*. Consider a roulette wheel, with numbers 00,0,1,2,3,...,36 (38 numbers in total). A ball is thrown onto the wheel as it is spinning, and comes to rest by one of the numbers. You always bet that the ball will stop on one of the numbers 1,2,...,12. Let N be the random variable giving the number of bets that you loose before your first win. Give pN (5) and FN (5).
4*. Consider a discrete random variable Y with Y (Ω) = { 0 , 1 ,... , 8 } and probability mass function pY given by
pY (k) =
c a if k = 0, 1 , 2 , 3 , 4 , 5 c a^2 if k = 6, 7 c(1 − a)^2 if k = 8,
where a is some fixed number between 0 and 1 and c is a constant to be determined. What value of c makes pY a mass function? (The answer is a function of a.)
5*. Let Y ∼ Pois(2). What is P (Y ≥ 2)?
6*. Let X be a continuous random variable that takes values in [0, 6] and whose distribu- tion function FX satisfies
FX (x) =
− 2 x^3 + 6x^2 + 144x 648
for 0 ≤ x ≤ 6.
Compute P
2 < X^ ≤^1
Give the probability density function of X in the interval [0, 6]?
7 †. Consider the experiment consisting of throwing a die twice, so the sample space is as in Example 3.1., Ω = {(ω 1 , ω 2 )|ω 1 , ω 2 ∈ { 1 , 2 ,... , 6 }}. Let D be the random variable giving the difference between the outcome of the first throw and the outcome of the second throw, D((ω 1 , ω 2 )) = ω 1 −ω 2. Sketch the graph of the probability mass function and the graph of the distribution function of D.
8 †. Consider the probability space corresponding to throwing a fair die. Give an example of a random variable X whose image is { 1 , 2 , 3 } and for which FX (π/2) = 1/2.
- Consider a discrete random variable V taking values in { 0 , 1 , 2 , 3 , 4 } with mass function pV given by
pV (x) =
(x + 1)cx^ if x = 0, 1 , 2 c(5 − x) if x = 3, 4. What value of c makes pV a mass function?
Introduction to Probability & Statistics 2019
- Let X ∼ Ber(p). Let Y = 1 − X and V = X^2. Show that
(a) Y ∼ Ber(1 − p); (b) V ∼ Ber(p).
- Alan and Bryan decide to duel, using just one six-shot revolver, and one bullet, between them. They decide to duel in the following way: with the bullet inserted into the revolver, Alan will spin the cylinder and shoot at Bryan (killing him if the gun fires); if he misses, Bryan will spin the cylinder and shoot at Alan. Assuming there is a 1/ 6 probability that the revolver will fire each time (since the revolver has 6 chambers), what is
(a) the distribution of the number of turns T until the gun fires? (b) the probability that Alan wins the duel?
- (P´olya’s Urn) At time 0, an urn contains 1 Red and 1 Black ball. Just before each time n = 1, 2 , 3 ,... , a ball is chosen at random from the urn and then replaced, along with a new ball of the same colour. For n = 0, 1 , 2 ,... and 1 ≤ r ≤ n + 1, let
pn,r = P (at time n, the urn contains exactly r Red balls).
(a) Write down the values of p 0 , 1 , p 1 , 1 and p 1 , 2 ; (b) By conditioning on the possible numbers of red balls at time 1, calculate p 2 ,r for r = 1, 2 , 3; (c) Can you find (and prove!) a general formula for pn,r? [HINT: guess at the right answer using your findings from parts (a) and (b). (Calculate p 3 ,r explicitly if you can’t see a pattern.) Then use induction.]
- A shop receives a batch of 1000 cheap lamps. The odds that a lamp is defective are 0 .1%. Let X be the number of defective lamps in the batch.
(a) What kind of distribution does X have? What is/are the value(s) of the param- eter(s) of this distribution? (b) What is the probability that the batch contains no defective lamps? One defective lamp? More than two defective lamps?
- You decide to play monthly in two different lotteries, and you stop playing as soon as you win a prize in one (or both) lotteries of at least one million euros. Suppose that every time you participate in these lotteries, the probability to win one million (or more) euros is p 1 for one of the lotteries and p 2 for the other. Let M be the number of times you participate in these lotteries until winning at least one prize. What kind of distribution does M have, and what is its parameter?
