Let's analyze the given cumulative distribution function [tex]\(F(x)\)[/tex] of the random variable [tex]\(X\)[/tex] and determine its values at various specified points. The function is defined as:
[tex]\[
F(x) =
\begin{cases}
0 & \text{if } x < 0 \\
0.2x & \text{if } 0 \leq x < 5 \\
1 & \text{if } 5 \leq x
\end{cases}
\][/tex]
We'll determine [tex]\( F(x) \)[/tex] at [tex]\( x = -1, 0, 2.5, 5, \)[/tex] and [tex]\( 6 \)[/tex].
1. For [tex]\( x = -1 \)[/tex]:
- Since [tex]\( -1 < 0 \)[/tex], we use the first case of the function.
[tex]\[
F(-1) = 0
\][/tex]
2. For [tex]\( x = 0 \)[/tex]:
- Since [tex]\( 0 \leq 0 < 5 \)[/tex], we use the second case of the function.
[tex]\[
F(0) = 0.2 \cdot 0 = 0
\][/tex]
3. For [tex]\( x = 2.5 \)[/tex]:
- Since [tex]\( 0 \leq 2.5 < 5 \)[/tex], we use the second case of the function.
[tex]\[
F(2.5) = 0.2 \cdot 2.5 = 0.5
\][/tex]
4. For [tex]\( x = 5 \)[/tex]:
- Since [tex]\( 5 \leq x \)[/tex], we use the third case of the function.
[tex]\[
F(5) = 1
\][/tex]
5. For [tex]\( x = 6 \)[/tex]:
- Since [tex]\( 6 \geq 5 \)[/tex], we use the third case of the function.
[tex]\[
F(6) = 1
\][/tex]
In summary, the values of the cumulative distribution function at the specified points are:
[tex]\[
\left( F(-1), F(0), F(2.5), F(5), F(6) \right) = (0, 0, 0.5, 1, 1)
\][/tex]
