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:

$E(X) =\int_{4}^7 \frac{1}{3} x$

$E(X) = \frac{1}{6} (7^2 -4^2) = 5.5$

Now we can find the second moment with this formula:

$E(X^2) =\int_{4}^7 \frac{1}{3} x^2$

$E(X^2) = \frac{1}{9} (7^3 -4^3) = 31$

And the variance for this case would be:

$Var(X)= E(X^2) -[E(X)]^2 = 31 -(5.5)^2 = 0.75$

And the standard deviation is:

$Sd(X)= \sqrt{0.75}= 0.866$

Step-by-step explanation:

For this case we have the following probability density function

$f(x)= \frac{1}{3}, 4 \leq x \leq 7$

And for this case we can find the expected value with this formula:

$E(X) =\int_{4}^7 \frac{1}{3} x$

$E(X) = \frac{1}{6} (7^2 -4^2) = 5.5$

Now we can find the second moment with this formula:

$E(X^2) =\int_{4}^7 \frac{1}{3} x^2$

$E(X^2) = \frac{1}{9} (7^3 -4^3) = 31$

And the variance for this case would be:

$Var(X)= E(X^2) -[E(X)]^2 = 31 -(5.5)^2 = 0.75$

And the standard deviation is:

$Sd(X)= \sqrt{0.75}= 0.866$