In this exercise it is necessary to use probability to calculate mean, variance and standard deviation; so we have:
a) [tex]1.75[/tex]
b) [tex]0.1875[/tex]
c) [tex]0.4330[/tex]
To calculate the mean is necessary to do the following steps:
[tex]E(x) = \in x; \rho(x)\\E(x) = (1)(0.25) + (2)(0.75)\\E(x) = 0.25 + 1.5\\E(x) = 1.75[/tex]
Now, to find the variance is needed to calculate:
[tex]E(x^2) - [E(x)]^2\\E(x^2) = \in x^2 * \rho(x)\\E(x^2) = (1^2)(0.25) + (2^2)(0,75)\\E(x^2) = (1)(o.25) + (4)(0.75)\\E(x^2) = 0.25 + 3\\E(x^2) = 3.25[/tex]
Then, subtract from the mean:
[tex]3.25 - (1.75)^2\\3.25 - 3.0625 = 0.1875[/tex]
To find the standard deviation it is necessary to do the square root of the variance:
[tex]\sqrt{Var(x)}\\\sqrt{0.1875} = 0.4330[/tex]
The standard deviation is needed to find PDF and CDF.
Probability Density Function (PDF): Using Microsoft excel:
When [tex]x = 1[/tex], mean = [tex](1.75,5)(\Delta) = 0.4330[/tex]
NORMADIST [tex](1, 1.75, 0.4330, FALSE) = 0.205562[/tex]
When [tex]x = 2[/tex], mean = [tex](1.75,5)(\Delta) = 0.4330[/tex]
NORMADIST [tex](1, 1.75, 0.4330, FALSE) = 0.779894[/tex]
Cumulative Density Function (CDF): Using Microsoft excel:
When [tex]x = 1[/tex], mean = [tex](1.75,5)(\Delta) = 0.4330[/tex]
NORMADIST [tex](1, 1.75, 0.4330, TRUE) = 0.041628[/tex]
When [tex]x = 2[/tex], mean = [tex](1.75,5)(\Delta) = 0.4330[/tex]
NORMADIST [tex](1, 1.75, 0.4330, TRUE) = 0.718154[/tex]
See more about probability at brainly.com/question/795909
