Interpretation of probability density greater than one

density functionmarginal-distributionmarginal-probabilityprobabilitystatistics

Good morning to everyone,

I'm trying to figure out how to interpret a probability density greater than one, but I didn't find any explanation that I considered satisfactory… Let me formalize my question.

Suppose that you have a random variable $p$ uniformly distributed over the support $[0,\frac 1 2]$. Obviously, it the probability density of this random variable is $f(p)=2$. Does this mean that if I marginally increase the realization of my random variable, the amount of probability "clustered around that new realization" (I couldn't find a better term to express the concept, I'm thinking about the graph of the cumulative, which has slope $2$ in this case) doubles?.

Sorry if I'm not being clear, I couldn't find better words to express my doubts.

Best Answer

The density function of a continuous random variable is not an uncountably infinite 'list' of probabilities. A continuous random variable has no probability at any one point.

A continuous random variable has positive probabilities only for intervals. (Intervals can be very short, but they cannot shrink of length $0.)$ The density function of a random variable $X$ provides a way to find probabilities such as $P(0 < X < 0.1).$ By convention, one writes $P(X = 0.0300)=0,$ and similarly for any other individual value.

Example 1. Let $X \sim \mathsf{Unif}(-.2, .2),$ with density function $f_X(t) = 2.5,$ for $-.2 \le x \le 2$ and $0$ elsewhere.

The total area under the density curve is $1.$ In order for that to be true, notice that the height of the density function must exceed $1$ for some values of $t.$ (See plot below.)

Then $$P(0 < X < 0.1) = \int_0^{0.1} f_X(t)\,dt = \int_0^{0.1} 2.5\, dt = 0.25.$$

Consider the plot below:

curve(dunif(x,-.2,.2), -.5,.5, col="blue",
      lwd=2, n=10001, ylab="PDF", xlab="t", 
      main="Density of UNIF(-.2, .2)")
 abline(h=0, col="green2")
 abline(v = c(0, .1), col="red")

The desired probability is the area beneath the (blue) density curve between the red vertical lines.

enter image description here

Example 2. Suppose that $Y \sim \mathsf{Norm}(\mu = 10, \sigma=0.1),$ with density function $f_Y(t).$ Then $P(Y \le 9.8) = \int_{-\infty}^{9.8} f_Y(t)\, dt = 0.02275.$

However, the this integral cannot be evaluated using the ordinary methods of calculus. One must use numerical integration or printed tables (obtained by numerical integration). In R, one can evaluate this integral using a normal cumulative distribution function (CDF) pnorm, (also obtained by numerical means).

pnorm(9.8, 10, .1)
[1] 0.02275013

In the figure below, the desired probability is the area under the density curve to the left of the vertical red line.

enter image description here

curve(dnorm(x, 10,.1), 9.5,10.5, col="blue",
      ylab="PDF", xlab="t", 
      main="Density of NORM(10, .1)")
 abline(h=0, col="green2")
 abline(v = 9.8, col="red")
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