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Question 6
Set up the dataframe.
(sim1 <- data.frame(trials = 1:10000,y = rbeta(10000,1,2))
%>%mutate(U = map_dbl(y,\(y) 1-2*y),V = map_dbl(y,\(y) 1/y),W = map_dbl(y,\(y) (y)^(1/2)))
) %>% head()
trials y U V W
1 1 0.1229633 0.7540733 8.132505 0.3506613
2 2 0.3680882 0.2638237 2.716740 0.6067027
3 3 0.3591177 0.2817645 2.784602 0.5992643
4 4 0.0235536 0.9528928 42.456354 0.1534718
5 5 0.1837906 0.6324188 5.440975 0.4287081
6 6 0.7089459 -0.4178918 1.410545 0.8419893
Plot simulated versus analytic for \(U=1-2Y\).
(ggplot(aes(x=U),data=sim1)
+ geom_histogram(aes(y =after_stat(density)),
fill = 'red',
binwidth = 0.05, center = 0.05)
+ geom_line(aes(y = (1+U)/2), col = 'blue', linewidth=1)
+ labs(y='P(U)', x= 'U')
+ theme_classic(base_size = 12)
)
Plot the simulated versus analytic for \(V=\frac{1}{Y}\).
(ggplot(aes(x=V),data=sim1)
+ geom_histogram(aes(y =after_stat(density)),
fill = 'red',
binwidth = 0.05, center = 0.05)
+ geom_line(aes(y = (2*V-2)/(V)^(3)), col = 'blue', linewidth=1)
+ labs(y='P(V)', x= 'V')
+ theme_classic(base_size = 12)
+ scale_x_continuous(limits = c(1,20))
)
Warning: Removed 957 rows containing non-finite outside the scale range
(`stat_bin()`).
Warning: Removed 2 rows containing missing values or values outside the scale range
(`geom_bar()`).
Warning: Removed 957 rows containing missing values or values outside the scale range
(`geom_line()`).
Plot simulated versus analytic \(W = \sqrt{Y}\).
(ggplot(aes(x=W),data=sim1)
+ geom_histogram(aes(y =after_stat(density)),
fill = 'red',
binwidth = 0.05, center = 0.05)
+ geom_line(aes(y = 4*W*(1-W^2)), col = 'blue', linewidth=1)
+ labs(y='P(W)', x= 'W')
+ theme_classic(base_size = 12)
)
