Question 7.2: Let X have a uniform distribution in the interval [𝛼, 𝛽]. Wh......
Let X have a uniform distribution in the interval [𝛼, 𝛽]. What is the distribution of the linear transformation
Y = 𝛾X + 𝛿
for 𝛾 > 0 and 𝛿 ∈ ℝ?
It is easier to work with distribution functions rather than densities. The distribution function of X is
F_X(t)= P(X \leq t)= \begin{cases} 0, \qquad ~ t < \alpha ,\\ \frac{t-\alpha}{\beta – \alpha }, \quad \alpha ≤ t ≤ \beta,\\ 1, \qquad ~ t > \beta. \end{cases}Before we consider the distribution function of Y, we first find its range of values. Since X ∼ \mathcal{V}[α,β], we have 𝛼 ≤ X ≤ 𝛽 and so, since 𝛾 is positive,
\gamma\alpha+\delta\leq Y=\gamma X+\delta\leq\gamma\beta+\delta(formally, we should say that this statement holds with probability one). Next, for y in the interval R_Y = [𝛾𝛼 + 𝛿, 𝛾𝛽 + 𝛿], we have
P(Y\leq y)=P(\gamma X+\delta\leq y)=P\left(X\leq{\frac{y-\delta}{Y}}\right),which, by the formula for F_X given above (replacing t by (y − 𝛿)∕𝛾 there), equals
{\frac{(y-\delta)/\gamma-\alpha}{\beta-\alpha}}={\frac{y-\delta-\gamma\alpha}{\gamma(\beta-\alpha)}}={\frac{y-(\delta+\gamma\alpha)}{\gamma(\beta-\alpha)}}.This gives the distribution function of Y as
F_Y(y)= P(Y \leq y)= \begin{cases} 0,~~~~~~~~~~~~~~~~~~~y\lt \gamma\alpha+\delta, \\ {\frac{y-(\delta+\gamma\alpha)}{\gamma(\beta-\alpha)}},\quad ~~~\gamma\alpha+\delta\leq y\leq\gamma\beta+\delta,\\ 1,\qquad\qquad\quad y\gt \gamma\beta+\delta. \end{cases}It is now easy to see (just put a = 𝛼𝛾 + 𝛿 and b = 𝛽𝛾 + 𝛿 in (7.2))
F(t)= \begin{cases} 0, \qquad t \lt a,\\ \frac{t-a}{b-a}, \quad a \leq t \leq b,\\ 1, \qquad t \gt b, \end{cases} (7.2)
that this is the distribution function of the uniform distribution over the interval [a, b]. In words, we have shown that a linear transformation of an uniformly distributed random variable also has an uniform distribution. More precisely, we have
X \sim \mathcal V[\alpha,\beta]\ \,\Rightarrow\,\gamma X+\delta \sim \mathcal V[\gamma\alpha+\delta,\gamma\beta+\delta].As a special case, assume that X ∼ \mathcal{V}[α,β] and consider the case when
\gamma={\frac{1}{\beta-\alpha}},\quad\delta=-{\frac{\alpha}{\beta-\alpha}}.Then, it follows that Y = 𝛾X + 𝛿 is uniformly distributed in the unit interval, [0, 1]. This distribution is commonly referred to as the standard uniform distribution.