In this paper, you need to do a Mathematical exploration. The report is on golden ratio; In which you link geometry and 2-3 other maths topics for a good exploration. Use the IB mathematics book for that. I have attached examples you may use same the concept but don’t copy. You need to have a…

(Auxiliary Functions and Moments). For a random variable X, the characteristic function E(eitX) always exists since eitX is a bounded random variable (with complex values). The moment generating function E(etx) only exists if X has moments of all orders and those moments do not grow too fast. a)      If Z is Gaussian with mean 0 and…

(A Box Calculus Verification). The purpose of this exercise is to generalize and unify the calculations we made for functions of Brownian motion with drift and geometric Brownian motion. It provides a proof of the validity of the box calculus for processes that are functions of Brownian motion and time. A)     Let Xt = f(t, Bt),…

Apply Student t regression (Section 5.7) to the stack loss data in Example 4.4, with degrees of freedom ν an unknown. Lange et al. (1989) consider these data under normal linear regression and Student regression and show support for the latter. In fact they report an estimate ν = 1.1. data, also much analysed, illustrate…

In Example 6.1 use a likelihood calculation and derive the posterior mean of the likelihood and deviance. Use the AIC and BIC criteria to compare solutions C = 1, 2, 3, 4. In Example 6.1 obtain the posterior probabilities under C = 3 that individual cases belong to different groups. These are averages over iterations…

In Example 7.5 (attitudes to working mothers) compare inferences from the residuals Wi − Xiβ with those based on Monte Carlo estimates of the conditional predictive ordinates (harmonic means of the sampled normal likelihoods for each subject). In Example 7.5 apply an ordered logistic model by data augmentation by direct sampling from a logistic and by…

In Example 11.6 (Indonesian rice farm data) assess gain from introducing AR1 errors (in addition to unstructured errors) in both random and fixed effects bi models. Also find the posterior probabilities that farms 1 to 171 are the best – in terms of having highest bi after allowing for inputs. Which farm has the highest probability of…

In Example 13.2 compare a 5-point discrete mixture on the log-logistic shape parameter with the variable scale model to downweight aberrant cases, namely ui ∼ L(ηi, 1/(κθi)) where θi are gamma with mean 1, and ui = log(ti). Commuter delay in work-to-home trips Washington et al. (2003) consider the durations of delay in work-to-home trips for 96 Seattle…

Suppose a binary response has true prevalence Pr(Y = 1) = π but that observed responses are subject to misclassification with probabilities α0 = Pr(y = 1|Y = 0), and α1 = Pr(y = 0|Y = 1). Assuming α0 = α1 = α, state the total probability P(yi = 1) in terms of the true prevalence probabilities P(Y = 1)…

Project: Section 1.3 focuses on using unlabeled, unrooted trees to model RNA secondary structure. However, rooted and/or labeled trees can also be used to model important aspects of RNA secondary structure. Explore this idea and how it affects the estimates of the number of possible RNA secondary structure techniques. As part of this exploration, consider…