1. Let the random variable X indicate the presence of a premature birth in San Diego, which happens for 10% of all child births. Let Y indicate the presence of ischemia in this child, and let the random variable Z represent the presence of a seizure. If a premature birth occurs, then a baby suffers from ischemia but not seizures with probability 0.2, seizures but not ischemia with probability 0.3, and both with probability 0.4. If a premature birth does not occur, then seizures and ischemia are statistically independent, where ischemia occurs with probability 0.1 and seizures occur with probability 0.2.
(a) If a baby is born prematurely, what is the probability neither a seizure nor ischemia occurs?
(b) Draw a tree diagram for this problem setup
(c) Calculate the probability a seizure occurs
(d) Calculate the probability a premature birth occurs, given that the baby has ischemia
(e) Assuming that births in San Diego occur independently with identical statistics, about how many births do we expect to see until the first baby is born with ischemia and no seizure?
2. let X be exponentially distributed with rate λX. Given X = u, let Y be a Poisson random variable of rate u.
(a) Calculate the expression for fX|Y (u|0) in closed form and draw it as a function of u (hint: try not to calculate any integrals).
(b) Give the expected value and variance of X given Y = 0.
(c) Are the random variables X and Y statistically independent?
3.
(a) Show (perhaps with an example) that for any three numbers (a, b, c) the event C = {min(a, b) > c} holds if and only if A = {a > c} and B = {b > c} hold. In other words, show that: C = A ∩ B.
(b) suppose X is an exponentially distributed random variable with rate λX and Y is an exponential random variable with rate λY . Calculate P (X > c) and P (Y > c).
(c) let X and Y from part (b) be statistically independent and Z be the random variable given by Z = min(X, Y ). Show that Z is exponentially distributed with rate λX + λY
4. Suppose we have a collection of Y1, . . . , Yn independent random variables that are exponentially distributed with fixed rate λ.
(a) Develop the maximum-likelihood estimate of θ , 1 λ and express the ML estimate ˆθ in terms of Sn = Σn i=1 Yi (hint: rst describe fY (v; θ) and then the log likelihood).
(b) Suppose we know up front that θ < 1. Give a 99% confidence interval on the ML estimate ˆθ for a fixed n. (hint: rst express the expectation and variance of ˆθ in terms of θ).
5. Suppose that Y is exponentially distributed with unknown rate λ0 or λ1, where 0 < λ0 < λ1.
(a) Show that the log likelihood ratio (LLR) hypothesis test with threshold τ is given by
(b) Give closed-form expressions (i.e. with no integrals) for the probability of type-I error and type-II errors for the LLR test with threshold τ .
6. Let a, b, c, d be integers where a ≤ b. Let N(a, b) be the number of heads counted in coin flips starting on trial a and concluding on trial b, where each coin flip is independent with P (heads) = p.
(a) Define the random variable X = N(a, b); calculate the PMF on X.
(b) Assume b < c ≤ d and define Y = N(c, d); explain why X and Y are statistically independent.
(c) Now, consider the case where b is not smaller than c and so X and Y are not necessarily independent. Let a = 1, b = 4, c = 4, d = 6 and calculate PX,Y (3, 2) (hint: use the total probability theorem).
7. Consider a hypothesis testing problem where under the null hypothesis, our observations Y1, . . . , Yn are independent and Gaussian with mean μ0 = 0 and variance σ2 = 1 2 . Under the alternate hypothesis, our observations are independent and Gaussian with mean μ1 = 10 and variance σ2 = 1 2 .
(a) Let n = 1. Suppose we observe the event E = { Y1 ∈ [3, 3 + 10-6]}.
Provide an approximation to Pθ0 (E), the probability of the event E under the null hypothesis, and Pθ1 (E), the probability of the event E under the alternate hypothesis.
(b) Provide a bound on the experiments n we must perform to guarantee that the probability of type-I and type-II errors are both below 0.01 using the ML rule. (hint: use the bound 1 - Φ(x) ≤ 1 2 e-x2 2 , where Φ is the CDF of a zero mean, unit variance Gaussian).