# # # Figure caption: The effect of the transformation y = a + bx # operating on a normally distributed random variable X having # mean muX and standard deviation sigmaX. The random variable Y = # a + bX is again normally distributed, with mean muY = a + b*muX # and standard deviation sigmaY = |b| sigmaX. The normal # distributions are displayed on the x and y axes; the linear # transformation is displayed as a line, which passes through the # point (muX, muY) so that it may be written, equivalently, as y - # muY = b(x - muX). # # For Y = a + b*X, we use a = 0, b = 0.7. a <- 0 b <- 0.7 # Set up the normal random variable X. x.mu <- 3 x.sd <- 0.35 # Set up Y as the transformation of X. y.mu <- b * x.mu + a y.sd <- abs(b) * x.sd x.max <- 2 * x.mu # Used for setting x-axes. # Get a reasonable range of values and evaluate the PDFs. x.values <- seq(from = x.mu - 4 * x.sd, to = x.mu + 4 * x.sd, by = 0.01) y.values <- a + b * x.values par(xaxs = "i") par(yaxs = "i") par(oma = rep(1, 4)) # Open an empty plot and set some axes. plot(0, 0, type = "n", xlim = c(0, 5), ylim = c(0, 3.5), xaxt = "n", xlab = "", yaxt = "n", ylab = "", main = "", bty = "n") # Add lines for X and Y's PDFs. lines(x.values, dnorm(x.values, x.mu, x.sd) * 0.8, lwd = 1.5) lines(dnorm(y.values, y.mu, y.sd) * 0.8, y.values, lwd = 1.5) # Add a line showing the relationship between X and Y. lines(c(0, 5), c(0, a+b*5), lwd = 2) # Add lines from the means of X and Y to the line showing their relationship. lines(c(x.mu, x.mu), c(0, y.mu), col = "gray", lty = 2, lwd = 2) lines(c(0, x.mu), c(y.mu, y.mu), col = "gray", lty = 2, lwd = 2) # Set labels. text(3.3, 3, expression(y-mu[y] == b(x-mu[x])), cex = 1.4) # (expression(phanton() ) is empty space) title(xlab = expression(paste(phantom(xxxxxxxxxx), mu[x])), line=0.5, cex.lab=1.4) title(ylab = expression(paste(phantom(xxxxxxxxx), mu[y])), line=0.2, cex.lab=1.4) # Close the graphics device png("figure9.1.png")