Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the firs...
6. Let S(t) denote the price of a security at time t. A popular model for the process S(t): t 0} supposes that the price remains unchanged until a "shock" occurs, at which time the price is multiplied by a random factor. If we let N(t) denote the number of shocks by time t, and let Xi denote the ith multiplicative factor then this model supposes that N (t) "x II (0)s ()s 6. Let S(t) denote the price of...
Let Xi , X2,. … X, denote a random sample of size n > 1 from a distribution with pdf f(x:0)--x'e®, x > 0 and θ > 0. a. Find the MLE for 0 b. Is the MLE unbiased? Show your steps. c. Find a complete sufficient statistic for 0. d. Find the UMVUE for θ. Make sure you indicate how you know it is the UMVUE. Let Xi , X2,. … X, denote a random sample of size n...
Exercise 1. Customers at a coffee shop ask for hot, denote it X 0, or cold, denote it Xn 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wants the same kind as their predecessor. (1) Find the PMF of N (2) What is the probability that XN+1-1? (3) Cold drinks take 30 seconds to prepare and hot drinks take 60 seconds. What is the expected time taken to serve...
python coding please 1.2 Sum of the Independent Random Variables Consider a set of 'n random variables XI,Xy . . . Х,, . Let's define the random variable Y as the stinmation of all X, variables: A) For the case m 10 and Xis being independent uniform variables in the interval -0.5,0.5, generate 100,000 samples of Y. Use the discretization technique from the previous section for the [-5,5 interval and plot the pmf of Y B) Now increase m to...
The following table presents the probabilities for the number of times that a certain computer system will crash in the course of a week. Let A be the event that there are more than two crashes during the week, let B be the event that the system crashes at least once. Number of Crashes Probability 0.60 0.30 0.05 0.04 0.01 4 Find the sample space. Then find the subsets of the sample space that correspond to the events A and...
An emergency department research group is interested in developing a statistical model for the number of in-hospital deaths among patients presenting to the emergency department with heart failure. Let N be the number of patients presenting to the emergency department with heart failure in a given year and assume that this random variable follows a geometric distribution p(N = n) = theta(1-theta)^n-1 n=1,...,infinite 0<theta<1 Next, for the ith patient define the random variable Yi to be equal to 1 if...
Two balls are placed randomly into two boxes labeled as I and II. Let X denote the number of balls in box I and Y denote the number of occupied boxes. (a) Find the joint density function of X and Y. (b) Compute E(X) and the conditional expectation E(X|Y= 1).
1) Customers at a coffee shop ask for hot, say Xn 0, or cold, say X 1, beverages according to a Bernoulli process with parameter p. Let N denote the first time that a customer wants the same kind as their predecessor. (a) Find the pmf of N. (b) What is the probability that XN+ 1? (c) If cold drinks take 30 seconds to prepare and hot drinks take 60 seconds. What is the expected time taken to serve the...
R and θ E (0, 1), define For x otherwise Let X, X, be i.i.d. random variables with density fo, for some unknown 0 E (0,1) 2 points possible (graded, results hidden) Let a be the number of X, which are negative (X, < 0) and b be the number of Xi which are non-negative (X 0). (Note that the total number of samples is n a b and be careful not to mix up the roles of a and...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved? Suppose...