Choose a number [math]U[/math] from the interval [math][0,1][/math] with uniform distribution. Find the density for the random variables [math]Y = |U - 1/2|[/math].

• [[math]]f(y) = \begin{cases}2|y-1/2|, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}1/2, \, 0 \leq y \leq 2\\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2 - |y-1/2|, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2, \, 0 \leq y \leq 1/2 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2y, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: D

We have

[[math]] P(|U-1/2| \leq y) = P(U \in (1/2-y, 1/2 + y) = \begin{cases} 2y, \, 0 \leq y \leq 1/2 \\ 1, \, y \gt 1/2 \\ 0 \,\, \textrm{Otherwise} \end{cases} [[/math]]

Taking the derivative of the distribution above, we obtain the density

[[math]] f(y) = \begin{cases} 2, \, 0 \leq y \leq 1/2 \\ 0 \,\, \textrm{Otherwise} \end{cases} [[/math]]