17  Quantile Functions

Example 17.1 Recall Example 15.3 where the waiting time, measured continuously in hours, from now until the next earthquake (of any magnitude) occurs in southern CA is a continuous random variable \(X\) with an Exponential distribution with rate parameter 2. The cdf of \(X\) is

\[ F_X(x) = 1- e^{-2x}, \; x \ge0. \]

  1. Evaluate and interpret \(Q_X(0.25)\).

  2. Evaluate and interpret \(Q_X(0.5)\).

  3. Find the quantile function of \(X\).

  4. How can you use the quantile function to construct a spinner for \(X\) (like in Figure 15.1)?

  5. Let \(U\) be a random variable with a Uniform(0, 1) distribution, and let \(Y = -0.5\log(1 - U)\). What is the distribution of \(Y\)? Explain without doing any calculations.

Example 17.2 Let \(Q\) be the quantile function of a Normal(0, 1) distribution.

Use the empirical rule for Normal distributions (Section 7.4) to find the following values and identify how they are represented in the standard Normal spinner (Figure 7.3).

  1. Find \(Q(0.025)\).

  2. Find \(Q(0.16)\).

  3. Find \(Q(0.25)\).

  4. Find \(Q(0.5)\).

  5. Find \(Q(0.75)\).

  6. Find \(Q(0.84)\).

  7. Find \(Q(0.975)\).

  1. If the cdf is a continuous, strictly increasing function over the range of possible values, then the quantile function is the inverse cdf. But the inverse of a cdf might not exist, if the cdf has jumps or flat spots. In particular, the inverse cdf does not exist for discrete random variables. So in general, the quantile function corresponding to cdf \(F\) is defined as \(Q(p) = \inf\{u:F(u)\ge p\}\).↩︎