Answer:
a. Find the probability that X is greater than 1: _<u>P(X>1) = 0.25</u>
b. Find the probability that X is less than .5: _<u>P(X<0.5)</u>_
c. Find the probability that X is equal to 1.5: <u> P(X=1.5)= 0</u>
Step-by-step explanation:
Hello!
The following density function describes a random variable X. f(x) = 1 − (x /2) if 0<x<2 a. Find the probability that X is greater than 1 ________ b. Find the probability that X is less than .5. _________ c. Find the probability that X is equal to 1.5.
First step to calculate the asked probabilities is to integrate the density function.
f(x) = 1 − (x /2) if 0<x<2
![\int\limits^2_0 {(1- (\frac{x}{2}))} \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E2_0%20%7B%281-%20%28%5Cfrac%7Bx%7D%7B2%7D%29%29%7D%20%5C%2C%20dx)
![\int\limits^2_0 {1} \, dx - \frac{1}{2} \int\limits^2_0 {x} \, dx](https://tex.z-dn.net/?f=%5Cint%5Climits%5E2_0%20%7B1%7D%20%5C%2C%20dx%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%5E2_0%20%7Bx%7D%20%5C%2C%20dx)
Now you resolve both integrals:
![\int\limits^2_0 {1} \, dx = x](https://tex.z-dn.net/?f=%5Cint%5Climits%5E2_0%20%7B1%7D%20%5C%2C%20dx%20%3D%20x)
![\frac{1}{2} \int\limits^2_0 {x} \, dx = \frac{1}{2} * \frac{x^2}{2} = \frac{x^2}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%5Cint%5Climits%5E2_0%20%7Bx%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%2A%20%5Cfrac%7Bx%5E2%7D%7B2%7D%20%3D%20%5Cfrac%7Bx%5E2%7D%7B4%7D)
= ![x-\frac{x^2}{4}](https://tex.z-dn.net/?f=x-%5Cfrac%7Bx%5E2%7D%7B4%7D)
The cummulative distribution is:
0 for x ≤ 0
for 0 < x < 2
1 for x ≥ 2
a. Find the probability that X is greater than 1.
P(X>1) = 1 - P(X ≤ 1)
"1" is included in the interval "0 < x < 2", to calculate the probability you have to replace it with
and replace X with 1
1 - P(X ≤ 1) = 1 - (
)= 1 - 075= 0.25
b. Find the probability that X is less than 0.5.
"0.5" in included in the interval "0 < x < 2", to calculate the probability you have to replace it with
and replace X with 0.5
P(X<0.5)=
= 0.4375
c. Find the probability that X is equal to 1.5.
"1.5" is included in the interval "0 < x < 2", to calculate the probability you have to replace it with
and replace X with 1.5
This is a continuous variable, in this type of variable the cumulative probability of X=k (k= constant) is always cero.
You can prove it doing the following calculation:
=
- (
) = 0
I hope it helps!