Sampling Distributions

---

Chi-squared

> # Generate 5 standard normal variates
> set.seed(2024)
> x1 <- rnorm(1000)
> x2 <- rnorm(1000)
> x3 <- rnorm(1000)
> x4 <- rnorm(1000)
> x5 <- rnorm(1000)
> 
> # Square the values to create chi-square distributed data of 5 degrees of freedom
> x_sq <- x1^2 + x2^2 + x3^2 + x4^2 + x5^2
> 
> # Plotting the chi-square distribution
> hist(x_sq, breaks = 20, freq = FALSE, 
+      main = "Chi-Square Distribution (df=5)", 
+      xlab = "Value", ylab = "Density", col = "skyblue", border = "white")
> curve(dchisq(x, df = 5), add = TRUE, col = "red", lwd = 2, lty = 2)
> legend("topright", legend = "Chi-Square Density", col = "red", lty = 2, bty = "n")

> # Generate chi-square distributed data
> degrees_freedom <- 5  # Degrees of freedom for chi-square distribution
> num_samples <- 1000   # Number of samples to generate
> chi_square_data <- rchisq(num_samples, degrees_freedom)
> 
> # Plotting the chi-square distribution
> hist(chi_square_data, breaks = 20, freq = FALSE, 
+      main = "Chi-Square Distribution (df=5) using built in function", xlab = "Value", ylab = "Density",
+      col = "skyblue", border = "white")
> curve(dchisq(x, degrees_freedom), add = TRUE, col = "red", lwd = 2, lty = 2)
> legend("topright", legend = "Chi-Square Density", col = "red", lty = 2, bty = "n")

Using simstudy

> library(simstudy)
> 
> # creating definitions table (contains formula and/or link)
> def <- defData(varname = "x1", formula = 0, variance = 1, dist = "normal") # standard normal
> def <- defData(def, "chisq1df", formula = "x1^2", dist = "nonrandom")       # add computed column, hence nonrandom
> def <- defData(def, "x2", formula = 0, variance = 1, dist = "normal")      # another standard normal
> def <- defData(def, "chisq2df", 
+                formula = "(x1^2) + (x2^2)", dist = "nonrandom")            # another computed column
> 
> set.seed(2024)
> dt <- genData(1000, def)   # generate 1000 rows for each definitions and/or formula
> head(dt)

Key: <id>
      id         x1   chisq1df          x2  chisq2df
   <int>      <num>      <num>       <num>     <num>
1:     1  0.9819694 0.96426392 -0.05169614 0.9669364
2:     2  0.4687150 0.21969379 -0.49938784 0.4690820
3:     3 -0.1079713 0.01165781 -0.90253166 0.8262212
4:     4 -0.2128782 0.04531711 -0.37196309 0.1836737
5:     5  1.1580985 1.34119203 -0.18843330 1.3766991
6:     6  1.2923548 1.67018105 -0.73661845 2.2127878

Visualizing standard normal distributions.

> library(ggplot2)
> library(gridExtra)
> p1 <- ggplot(dt, aes(x = x1)) +
+   geom_histogram(aes(y = after_stat(density)), fill = "skyblue", color = "white") +
+   geom_density(fill = "skyblue", color = "red", alpha = 0.5) +
+   theme_bw() + labs(x = "X1")
> p2 <- ggplot(dt, aes(x = x2)) +
+   geom_histogram(aes(y = after_stat(density)), fill = "skyblue", color = "white") +
+   geom_density(fill = "skyblue", color = "red", alpha = 0.5) +
+   theme_bw() + labs(x = "X2")
> grid.arrange(p1, p2, nrow = 2)

> p1 <- ggplot(dt, aes(x = chisq1df)) +
+   geom_histogram(aes(y = after_stat(density)), fill = "royalblue", color = "white") +
+   geom_density(fill = "royalblue", color = "black", alpha = 0.5) + 
+   theme_bw() + labs(x = "Chi-squared distribition (df = 1)"); p1

> p2 <- ggplot(dt, aes(x = chisq2df)) +
+   geom_histogram(aes(y = after_stat(density)), fill = "royalblue", color = "white") +
+   geom_density(fill = "royalblue", color = "black", alpha = 0.5) + 
+   theme_bw() + labs(x = "Chi-squared distribition (df = 2)"); p2

> grid.arrange(p1, p2, nrow = 2)

Poisson

Mean and variance of Poisson variate are equal.

> def <- defData(varname = "h", formula = 40, dist = "poisson")   # mean = 40
> set.seed(2024)
> dt <- genData(1000, def)
> 
> round(dt[, .(avg = mean(h), var = var(h))], 2)

     avg   var
   <num> <num>
1: 39.71 37.84