Suppose that f(x) = 3 (8x - x^2)/256 for 0 < x < 8 Determine the following probabilities. Round all of your answers to 3 decimal places (e. g. 98.765).
a. P(X < 5)
b. P(X< 9)
c. P(5 < X < 7)
d. P(X > 3)
e. Determine x such that P(X < x) = 0.95

Answer and Step-by-step explanation: This function is a probability density function of a random variable X:

and to calculate probabilities, we will have to integrate it:

Solving:

Then:

a. P(X < 5)

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)

P(5 < X < 7) = 0.273

d. P(X > 3)

P (X > 3) = 0.683

e. P(X < x) = 0.95

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.

kaahe btau