In order to calculate the parking fee, mayor of the city is using a random variable $X 〜 Τ([0; 25]).$ The value of $X$ is used to calculate the parking fee $Y = max(X - 7, 0)$. Find the expected value and standard deviation of the parking fee.
My solution:
Since the random variable is $X 〜 Τ([0; 25])$, I can calculate the density $F_x = \frac{1}{25}$
In order to determine the expected value, I'm using the integral $$E(X) =\int_7^{25}((x-7) ⋅F_x)dx= \int_7^{25}((x-7) ⋅\frac{1}{25})dx=$$$$=\frac{1}{25}⋅\int_7^{25}(x-7)dx = \frac{1}{25} ⋅162 = 6.48$$
In order to determine the variance needed in order to calculate the standard deviation, I'm using the integral
$$D(X) =\int_7^{25}((x-7)^2 ⋅F_x)dx - (E(X))^2= \int_7^{25}((x-7)^2 ⋅\frac{1}{25})dx - (6.48)^2 =$$
$$=\frac{1}{25}⋅\int_7^{25}(x-7)^2dx - (6.48)^2 =\frac{1}{25} ⋅ 162 - 41.99 ≈ 35.77$$
In order to get the standard deviation, I'm getting the root of variance
$$\sqrt{D(X)} = \sqrt{(35.77)} ≈ 5.9808 $$
Is this correct? The numbers seem off to me.