Using integrals, the desired probabilities and measures are given as follows:
a) P(X < 6) = 0.84375 = 84.375%.
b) P(X < 13) = 1 = 100%.
c) P(6 < X < 8) = 0.15625 = 15.625%.
d) P(X > 5) = 0.31641 = 31.641%.
e) x = 6.434.
What is the probability distribution?
The distribution is given as follows, simplifying the operations:
f(x) = 0.09375x - 0.01171875x², 0 ≤ x ≤ 8.
Hence the density function is:
[tex]\int_{a}^{b} f(x) dx[/tex]
[tex]\int_{a}^{b} (0.09375x - 0.01171875x^2) dx[/tex]
[tex]0.046875x^2 - 0.00390625x^3|_{x=a}^{x = b}[/tex]
Applying the Fundamental Theorem of Calculus:
P(a < X < b) = 0.046875b² - 0.00390625b³ - 0.046875a² + 0.00390625a³
Hence, for item a:
P(X < 6) = 0.046875(6)² - 0.00390625(6)³.
P(X < 6) = 0.84375 = 84.375%.
For item b, the entire distribution is below 13, hence:
P(X < 13) = 1 = 100%.
For item c:
P(6 < X < 8) = 0.046875(8)² - 0.00390625(8)³ - 0.046875(6)² + 0.00390625(6)³.
P(6 < X < 8) = 0.15625 = 15.625%.
For item d:
P(5 < X < 8) = 0.046875(8)² - 0.00390625(8)³ - 0.046875(5)² + 0.00390625(5)³.
P(X > 5) = 0.31641 = 31.641%.
For item e:
We have to find b when a = 0 and P(X < b) = 0.9, hence:
0.046875b² - 0.00390625b³ = 0.9.
Using a calculator, the solution in the range of the distribution is given by:
b = x = 6.434.
More can be learned about integrals and probabilities at brainly.com/question/15109629
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