STA 601 Homework 1

Let \(\theta=\)proportion of the total points possible that you will receive in this class (e.g., 0.86 implies that you got an 86% and would have gotten an A- after the curve); Taking a subjective Bayes approach, choose a prior density for \(\theta\); Plot this density in R and provide some summaries.

The grade percentage, theta, ranges from 0 to 1. It is most likely for me to received A-, but less likely to receive A/A+, B+ and B. It is very unlikely to receive grade lower than B. Therefore, I can assume that the distribution of prior theta is a beta distribution that is negative skewed and concentrates on the very right side. The left and right parameters are \(a=30\) and \(b=4\). We can plot the prior density curve below:

a = 30
b=4
theta<-seq(0,1,length=500)

The expected grade proportion \(\theta\) is: \(\frac{a}{a+b}=0.88\); The most probable proportion is \(\frac{a-1}{a+b-2}=0.90625\). The median is 0.8898513.

The 95% interval of the proportion will be (0.7566836, 0.9659671).

(low = qbeta(0.025, a, b))

## [1] 0.7567

(high = qbeta(0.975, a, b))

## [1] 0.966

a=30
b=4

theta <- seq(0,1,length=500)

plot(theta,dbeta(theta,a, b), type="l", xlab = "percentage of grade I will receive",
      ylab = "Density", lwd = 2, ylim = c(0,10)     
)

abline(v = low, col = 4)
abline(v = high, col = 4)

title ("95% Interval for My Grade Proportion")