Probability (Uncertainty and Data)

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Unit 10: Markov chains

Oscar goes for a run each morning. When he leaves his house for his run, he is equally likely to use either the front or the back door; and similarly, when he returns, he is equally likely to use either the front or the back door. Assume that his choice of the door through which he leaves is independent of his choice of the door through which he returns, and also assume that these choices are independent across days.

Oscar owns only five pairs of running shoes, each pair placed at one of the two doors. If there is at least one pair of shoes at the door through which he leaves, he wears a pair for his run; otherwise, he runs barefoot. When he returns from his run, if he wore shoes for that run, he takes off the shoes after the run and leaves them at the door through which he returns.

We wish to determine the long-term proportion of time that Oscar runs barefoot.

Unit 9: Bernoulli and Poisson processes

Marie distributes toys for toddlers. She makes visits to households and gives away one toy only on visits for which the door is answered and a toddler is in residence. On any visit, the probability of the door being answered is , and the probability that there is a toddler in residence is . Assume that the events “Door answered" and “Toddler in residence" are independent and also that events related to different households are independent.

Unit 8: Limit theorems and classical statistics

For each of the following sequences, determine whether it converges in probability to a constant. If it does, enter the value of the limit. If it does not, enter the number “999".

Exam 2

Problem 1(a)
Suppose that X,Y, and Z are independent, with E[X]=E[Y]=E[Z]=2, and E[X^2 ]=E[Y^2 ]=E[Z^2 ]=5.
Find cov⁡(XY,XZ).

Unit 7: Bayesian inference

The lifetime of a type-A bulb is exponentially distributed with parameter λ. The lifetime of a type-B bulb is exponentially distributed with parameter μ, where μ>λ>0. You have a box full of lightbulbs of the same type, and you would like to know whether they are of type A or B. Assume an a priori probability of 1/4 that the box contains type-B lightbulbs.

Assume that μ≥3λ. You observe the value t_1 of the lifetime, T_1, of a lightbulb. A MAP decision rule decides that the lightbulb is of type A if and only if t_1≥α. Find α, and express your answer in terms of μ and λ. Use 'mu" and 'lambda" and 'In" to denote μ,λ, and the natural logarithm function, respectively.

Unit 6: Further topics on random variables

Let X and Y  be independent random variables, each uniformly distributed on the interval [0,1].

Unit 5: Continuous random variables

Sophia is vacationing in Monte Carlo. On any given night, she takes X dollars to the casino and returns with Y dollars. The random variable X has the PDF shown in the figure. Conditional on X=x, the continuous random variable Y is uniformly distributed between zero and 3x.

Exam 1

Amy immigrated to a new city, and would like to make friends with her new neighbours.
On any particular day i, she feels shy with probability 1-p(0<p<1) and stays home; or, with probability p, she goes out and visits the i th house in her neighborhood. At any house that she visits, either:
(i) someone is at the house and answers the door; this happens with probability q (where 0<q<1 ). In that case, Amy shows them a picture of her hometown;
(ii) no one is at the house and Amy returns home.
We assume that the collection of all events of the form {Amy stays home on day i} and { Someone is at the i-th house on day i}, for i=1,2,…, are (mutually) independent.

Unit 4: Discrete random variables

Consider a sequence of n+1 independent tosses of a biased coin, at times k=0,1,2,…,n. On each toss, the probability of Heads is p, and the probability of Tails is 1-p.
A reward of one unit is given at time k, for k∈{1,2,…,n}, if the toss at time k resulted in Tails and the toss at time k-1 resulted in Heads. Otherwise, no reward is given at time k.
Let R be the sum of the rewards collected at times 1,2,…,n.
We will find E[R] and Var⁡(R) by carrying out a sequence of steps. Express your answers below in terms of p and/or n using standard notation (available through the "STANDARD NOTATION" button below.) Remember to write "*" for all multiplications and to include parentheses where necessary.

Unit 3: Counting

Each one of n persons, indexed by 1,2, ... n, has a clean hat and throws it into a box. The persons then pick hats from the box, at random. Every assignment of the hats to the persons is equally likely. In an equivalent model, each person picks a hat, one at a time, in the order of their index, with each one of the remaining hats being equally likely to be picked. Find the probability of the following events.

Unit 2: Conditioning and independence

Before leaving for work, Serap checks the weather report in order to decide whether to carry an umbrella. On any given day, with probability 0.2  the forecast is “rain" and with probability 0.8 the forecast is “no rain". If the forecast is “rain", the probability of actually having rain on that day is 0.8. On the other hand, if the forecast is “no rain", the probability of actually raining is 0.1.

Unit 1: Probability models and axioms

Mary and Tom park their cars in an empty parking lot with n >= 2 consecutive parking spaces (i.e, n spaces in a row, where only one car fits in each space). Mary and Tom pick parking spaces at random; of course, they must each choose a different space. (All pairs of distinct parking spaces are equally likely.) What is the probability that there is at most one empty parking space between them?

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