Respuesta :
Answer and Step-by-step explanation: This function is a probability density function of a random variable X:
[tex]f(x)=\frac{3(8x-x^{2})}{256}[/tex]
and to calculate probabilities, we will have to integrate it:
[tex]P(X<x)=\int\limits^x_0 {\frac{3}{256}(8x-x^{2}) } \, dx[/tex]
Solving:
[tex]P(X<x)=\frac{3}{256}(4x^{2}-\frac{x^{3}}{3} )[/tex]
Then:
a. P(X < 5)
[tex]P(X<5)=\int\limits^5_0 {\frac{3}{256}(8x-x^{2}) } \, dx[/tex]
[tex]P(X<5)=\frac{3}{256}(100-\frac{125}{3} )[/tex]
[tex]P(X<5)=\frac{3}{256}(\frac{300-125}{3} )[/tex]
[tex]P(X<5)=\frac{175}{256}[/tex]
P(X < 5) = 0.683
b. P(X < 9)
Since density function's upper limit is 8, probability of x < 9 is 100% or 1.
So, P(X < 9) = 1
c. P(5 < X < 7)
[tex]P(5<X<7)=\int\limits^7_5 {\frac{3}{256}(8x-x^{2}) } \, dx[/tex]
[tex]P(5<X<7)=\frac{3}{256}[4(7)^{2}-\frac{7^{3}}{3}-4(5)^{2}+\frac{5^{3}}{3}][/tex]
[tex]P(5<X<7)=\frac{3}{256}[96-\frac{218}{3} ][/tex]
[tex]P(5<X<7)=\frac{70}{256}[/tex]
P(5 < X < 7) = 0.273
d. P(X > 3)
[tex]P(X>3)=\int\limits^8_3 {\frac{3}{256}(8x-x^{2}) } \, dx[/tex]
[tex]P(X>3)=\frac{3}{256}[256-\frac{512}{3}-36+9 ][/tex]
[tex]P(X>3)=\frac{3}{256}(229-\frac{512}{3} )[/tex]
[tex]P(X>3)=\frac{175}{256}[/tex]
P (X > 3) = 0.683
e. P(X < x) = 0.95
[tex]0.95=\int\limits^x_0 {\frac{3}{256}(8x-x^{2}) } \, dx[/tex]
[tex]\frac{3}{256}[\frac{12x^{2}-x^{3}}{3} ] =0.95[/tex]
[tex]\frac{12x^{2}-x^{3}}{256}=0.95[/tex]
[tex]12x^{2}-x^{3}=243.2[/tex]
[tex]-x^{3}+12x^{2}-243.2=0[/tex]
Solving this cubic equation, we have three values for x:
x₁ = -3.909
x₂ = 8.992
x₃ = 6.917
The value of x will the one between 0 and 8, which are the limits of the function. So, value of x which gives a probability of 0.95 is x = 6.917.
