Question

If $\mathrm{x}$is uniformly distributed over $(\mathrm{a},\mathrm{b})$, what random variable, having a linear relation with $\mathrm{x}$, is uniformly distributed over $(0,1)$$?$

Short answer

Random variable having linear relation with$\mathrm{x}$is$\mathrm{Y}=\frac{\mathrm{X}-\mathrm{a}}{\mathrm{b}-\mathrm{a}}$

Step-by-step solution

Random variable

Given that $\mathrm{X}~Unif(\mathrm{a},\mathrm{b})$.

Approach Random variable

$\mathrm{Y}=\frac{\mathrm{X}-\mathrm{a}}{\mathrm{b}-\mathrm{a}}$

Explanation

Since$\mathrm{X}\in (\mathrm{a},\mathrm{b})$.

$\mathrm{Y}\in (0,1)$.

For $\mathrm{y}\in (0,1)$,

$\mathrm{P}(\mathrm{Y}\le \mathrm{y})=\mathrm{P}\left(\frac{\mathrm{X}-\mathrm{a}}{\mathrm{b}-\mathrm{a}}\le \mathrm{y}\right)$

$=\mathrm{P}(\mathrm{X}\le \mathrm{a}+(\mathrm{b}-\mathrm{a}\left)\mathrm{y}\right)$

$=\frac{\mathrm{a}+(\mathrm{b}-\mathrm{a})\mathrm{y}-\mathrm{a}}{\mathrm{b}-\mathrm{a}}$

$=\mathrm{y}$

Proved that $\mathrm{Y}~Unif(0,1)$.