Definition 5.1.
The uniform distribution function on the interval \([a,b]\) is given by
\begin{equation}
F(x) =
\begin{cases}
0 \amp x\lt a\\
\frac{x-a}{b-a} \amp a\leq x\leq b\\
1 \amp x\geq b.
\end{cases}\tag{5.1}
\end{equation}
Section 5 Continuous Random Variables
Subsection 5.1 General distribution functions
It turns out that the properties given in Proposition 4.5 hold for all quantitative random variables, whether discrete or continuous, and no matter whether the underlying probability model is discrete or general. Even better, it turns out that if \(F\colon \R\to\R\) is any function that satisfies the three properties in Proposition 4.5, then there exists a probability model \((\Omega,P)\) and a random variable \(X\colon \Omega\to
\R\) such that \(F=F_X\text{.}\)
1
See [5], Theorem 1.5.3, p.31.
Distribution functions for discrete random variables are piecewise constant functions that are discontinuous. The properties in Proposition 4.5 allow for the possibility of continuous distribution functions. The simplest of these is the following.
Checkpoint 5.2.
-
Sketch the graph of the uniform distribution function.
-
Show that the uniform distribution function satisfies the three properties of Proposition 4.5.
-
Show that the uniform distribution function \(F\) defined in Definition 5.1 is the distribution function \(F_X\) for the variable \(X\colon \R\to\R\) given by \(X(\omega)=\omega\text{,}\) and where probability measure on \(\Omega=\R\) is the uniform probability measure (3.1) for subsets of \([a,b]\text{,}\) and the probability measure is zero for subsets outside of \([a,b]\text{.}\)
A random variable is called continuous if its distribution function is continuous on all of the real line. If the distribution function \(F_X\) of a random variable \(X\) is differentiable (or piecewise differentiable, with at most a finite or countably infinite number of points of nondifferentiability), then the derivative \(f_X = F_X'\) is called the probability density function for \(X\text{.}\) If \(X\) has a probability density function \(f_X\text{,}\) then we have, for all intervals \([a,b]\text{,}\)
\begin{equation}
P(X\in [a,b]) =
F_X(b)-F_X(a)=\int_a^b f_X(x)\; dx.\tag{5.2}
\end{equation}
Checkpoint 5.3.
-
Sketch the graph of the probability density function for the uniform distribution function.
-
Show that, if \(f\colon \R\to [0,\infty)\) is an integrable function with \(\int_{-\infty}^{\infty} f(x)\;dx = 1\text{,}\) then \(f\) is the probability density function of some random variable \(X\) with distribution function \(F_X\) that satisfies \(F_X'=f\text{.}\)
-
Justify (5.2).
If \(X\) is a continuous random variable with probability density function \(f_X\text{,}\) then \(E(X)\) is given by
\begin{equation}
E(X) = \int_{-\infty}^{\infty} xf_X(x)\;dx\tag{5.3}
\end{equation}
if the improper integral on the right converges. If both of the integrals \(\int_{-\infty}^{\infty} x f_X(x)\;dx,
\int_{-\infty}^{\infty} x^2 f_X(x)\;dx \) converge, then the variance of \(X\) is given by
\begin{align}
\var(X) \amp = E(X^2)-E(X)^2\tag{5.4}\\
\amp = \int_{-\infty}^{\infty} x^2
f_X(x)\;dx - \left(\int_{-\infty}^{\infty} x f_X(x)\;dx\right).\tag{5.5}
\end{align}
Checkpoint 5.4.
Explain how the formula (5.3) for expected value of a continuous random variable connects to expected value for a discrete variable.
Exercises 5.2 Exercises
1.
Let \(k,p\) be constants, and let \(f_{k,p}\colon \R\to
\R\) be given by
\begin{equation*}
f_{k,p}(x)=
\begin{cases}
k/x^p \amp x\geq 1\\
0 \amp x\lt 1
\end{cases}.
\end{equation*}
-
Find all values of \(k,p\) such that \(f_{k,p}\) is a density function.
-
For each density function \(f_{k,p}\text{,}\) find the corresponding distribution function \(F_{k,p}\text{.}\)
-
Let \(X_{k,p}\) be a random variable with distribution function \(F_{k,p}\text{.}\) Find all values of \(k,p\) such that \(E(X)\) exists. Find all values of \(k,p\) such that \(\var(X)\) exists.
2.
Let \(X,Y\) be continuous random variables, and suppose there exist intervals \(A=(a_1,a_2]\text{,}\) \(B=(b_1,b_2]\) such that \(P(X\in A \text{ and } Y\in B)\neq P(X\in A)P(Y\in B)\text{.}\) Show that \(X,Y\) are dependent variables. That is, show that \(F_{XY}\neq F_XF_Y\text{.}\)
