Respuesta :
Answer:
Following are the answer to this question:
Step-by-step explanation:
In the given equation there is mistyping so, correct equation and its calculation can be defined as follows:
Given:
[tex]f(x) =\left\begin{array}{cc} 0&x< 1\\0.5& 1 < x<3\\ 1&3 < x\end{array}\right[/tex]
Calculated value:
[tex]1) \ \ P(x < 3) =1\\\\2) \ \ P(x < 2) = 1-0.5 \\[/tex]
[tex]= 0.5[/tex]
[tex]3) P(1 < x < 2) = P( x< 2) -p(x< 1)\\\\[/tex]
[tex]=0.5-0\\=0.5\\[/tex]
[tex]4) \ \ P(x> 2)= 1-P(x<2)[/tex]
[tex]=1-0.5\\=0.5\\[/tex]
That's why the given equation is true.
The probabilities are:
[tex]1. \:P(x < 3) =1\\2.\: P(x < 2) = 0.5\\3.\: P(1 < x < 2) = 0.5\\4.\: P(x > 2) = 0.5[/tex]
Given function is:
[tex]\[ f(x)= \begin{cases} 0, \: \text{x}< 1\\ 0.5, \: 1 < x < 3\\1, \: x \geq 3\end{cases}\][/tex]
Verification that f(x) is Cumulative distribution function:
1. Since for x > 3, f(x) = 1, thus we have [tex]\lim_{x \to \infty} f(x) = 1[/tex]
2. Since for x < 1, f(x) = 0, thus we have [tex]\lim_{x \to -\infty} f(x) = 0[/tex]
3. Since values of f(x) are not decreasing as x is increasing, thus f(x) is non decreasing.
4. f(x) is right continuous too.
Thus f(x) is a cumulative distribution function.
The Probabilities are calculated as follows:
[tex]1. \:P(x < 3) = f(3) =1\\2.\: P(x < 2) = f(2) = 0.5\\3.\: P(1 < x < 2) = P(x < 2) - P(x < 1)= 0.5 - 0 = 0.5\\4. \: P(x > 2) = 1 - P(x < 2 \: and \: x = 2) = 1 - 0.5 = 0.5[/tex]
Learn more here:
https://brainly.com/question/19884447
