9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,...
COMPLETE TWO PROBLEMS FROM AMONG PROBLEMS 6-9 6. (20 pts) Consider the following hidden Markov model (HMM): . x = (x,,. . . ,x,Je {o, l}n [i.e., X is a binary sequence of length n] and Y = (Yİ, . . . , Y, ) E R" [i.e., Y is a sequence of n real numbers.] ·Xi ~ Bernoulli (1 /2) Ip is the switching probability; when p is small the Markov chain likes to stay in the same state]...
3. [20 Points] Assume that we have the Hidden Markov Model (HMM) depicted in the figure below [4 Points] If each of the states can take on k different values and a total of m a. possible (across all states), how many parameters are different observations are required to fully define this HMM? Justify your answer b. [4 Points] What conditional independences hold in this HMM? Justify your answer [12 Points] Suppose that we have binary states (labeled A and...
I just need some help with question part c. We first examine a simple hidden Markov model (HMM). We observe a sequence of rolls of a four-sided die at an "occasionally dishonest casino", where at time t the observed outcome x E {1, 2, 3,4}. At each of these times, the casino can be in one of two states zi E {1, 2}. When z,= 1 the casino uses a fair die, while when z,- 2 the die is biased...
REDIT (10 pts). Suppose X = (Xi,Xy, ,x,000) are random variables taking values S Xi S 1 for all 1). Design a hypothesis test that tests the null hypothesis that 1, X', ...X,oo are iid (independent and identically-distributed) and uniform on [0, 1] at significance level α. (Recall that the best way to do this is to i) choose a statistic S(X) that you can compute the tion of, assuming the null hypothesis is true, and ii) use that to...
Show all work and answer fully! will give a good rating for a good solution 2. Let {y,/-1 be a sequence of independent identically distributed random = k) = ak for variables with values in the set S {0,1, ali n E N and k E S. Let Xo = 0 and ,9), where P( x, = Yǐ + . . . + Y,, (mod 10). Show that XJn-o is a Markov chain, and find its transition probabilities in terms...
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qualifiable, hidden results) Assume we do not actually get to observe Xı , . . . , X. . Instead let Yı , . . . , Y, be our observations where Yi = 1 (Xi 0.5) . Our goal is to estimate 0 based on this new data. What distribution does Y follow? First, choose the type of distribution:...
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely....
5. (15 points) The shopping times of n = 64 randomly selected customers at a local supermarket were recorded. The average and variance of the 64 shopping times were 33 minutes and 356 minutes, respectively. Estimate u, the true average shopping time per customer, with a confidence coefficient of 1-a = 0.90. 6. (10 points) Let X1, X2, ..., Xn denote n independent and identically distributed Bernoulli random vari- ables s.t. P(X; = 1) = p and P(Xi = 0)...
please dont copy other answer, it is not correct model from before: Continue with the same network availability model from the previous problem. Engineers decided to pool two types of failures together considering the two-state Markov chain with the state space. S-(0,1), and infinitesimal transition probabilities defined (under the assumption that A 0) as follows. They set λ-λι + λ2 and μ-μι + μ2 and then considered the infinitesimal transition probabilities as P [x(t + Δ) = 1 lx(t) =...
In this problem, we will model the likelihood of a particular client of a financial firm defaulting on his or her loans based on previous transactions. There are only two outcomes, "Yes" or "No", depending on whether the client eventually defaults or not. It is believed that the client's current balance is a good predictor for this outcome, so that the more money is spent without paying, the more likely it is for that person to default. For each x,...