Question

Please HELP best answer will receive a BRAINLIEST. Given the probability density function f ( x ) = 1/3 over the interval [ 4 , 7 ] , find the expected value, the mean, the variance and the standard deviation.

Answer 1

Answer:

Step-by-step explanation:

Assume that f(x) = 0 for x outside the interval [4,7]. We will use the following

$E[X^k] = \int_{4}^{7}x^k f(x) dx$

$Var(X) = E[X^2]- (E[X])^2$

Standard deviation = $\sqrt[]{Var(X)}$

Mean = $E[X]$

Then,

$E[X] = \int_{4}^{7}\frac{1}{3}dx = \frac{7^2-4^2}{2\cdot 3} = \frac{11}{2}$

$E[X^2] = \int_{4}^{7}\frac{x^2}{3}dx = \frac{7^3-4^3}{3\cdot 3} = 31$

Then, $Var(x) = 31-(\frac{11}{2})^2 = \frac{3}{4}$

Then the standard deviation is $\frac{\sqrt[]{3}}{2}$