Identify which choices would be considered descriptive statistics and which would be considered inferential statistics.
a. Of 500 randomly selected people in New York City, 210 people had O+ blood.
b. "42 percent of the people in New York City have O+ blood." Is the statement descriptive statistics or inferential statistics?
c. "58 percent of the people of New York City do not have type O+ blood." Is the statement descriptive statistics or inferential statistics?
d. "42 percent of all people living in New York State have type O+ blood." Is the statement descriptive statistics or inferential statistics?
Consider a simple model of the hunting behaviour of a fox. In a single day, the fox catches either 0 or 1 rabbits, with probabilities 0.3 and 0.7, respectively. The outcome of hunting on day m is independent of all other days. We describe this system through a discrete-time, discrete-state stochastic process R(m), where R is the total number of rabbits caught between the end of day 0 (defining a starting point of our observation) and the end of day m. m = 0, 1, 2, 3... is thus a discrete time parameter, and R(0) =0 by definition. (a) Give a graphical illustration of the process R(m). (b) Calculate the probability distribution P( R = r, m) = p.(m) after m = 3 days. Hint: you can use a truncated transition matric to do this. (c) Working with the appropriate row of the transition matrix, write down the relatonship between po(m) and po(m - 1). Using this equation, derive an expression for the probability that the fox has caught no rabbits at all since day 0 by the end of day m. (d) Working with the appropriate row of the transition matrix, write down the relationship between p, (m), Po(m - 1) and pi(m - 1). Using this equation, and the solution to part (d), verify that Pi(m) = m x 0.7 x 0.3m-1 (1) Hint: you can substitute the candidate solution into your expression relating p(m), po(m-1) and p (m-1). just as if you were checking a candidate solution to a differential equation. Don't forget to check the initial conditions. (e) It can be shown that pr(m) - the probability of catching R = r rabbits in m days - is given by a binomial distribution (see Appendix A.2.1). Here, m is the number of "trials", r the number of successes and q = 0.7 the probability of success. Hence give expressions for: i The mean number of rabbits caught after in days. if The standard deviation of the total number rabbits caught after m days. ill The standard deviation of the mean number of rabbits caught per day after m days. Act (f) Write code to generate sample trajectories of this process. Run 1000 sample trajectories for 30 days and plot your estimates of the three quantities calculated in part (e) as a function of m. Go (g) Explain whether the binomial distribution in part (e) is well-approximated by a Poisson distribution for allConsider the case of Bayesian inference, i.e. minimizing L(S, Sbayes) P(sr) ds for the case L = (s - Sbayes)2. By differentiating wrt Sbayes, show the optimal estimate in this case is Sbayes = sP(s/r)dsConsider the same assumption in previous question. In addition, we consider a possible vomiting out of coughing as shown in the following network. where V = vomiting [v or -v). We assume that P(v|c) = 0.2 and P(v|-c) = 0.1. Use Bayesian exact inference to obtain the following probabilities: 1. Ptf,a,c,-v) 2. Plflv) Are the following statements true? Justify your answers. 1.F__A|V 2.F__V|A
