Answer :
Answer:
Step-by-step explanation:
Assume that f(x) = 0 for x outside the interval [4,7]. We will use the following
[tex]E[X^k] = \int_{4}^{7}x^k f(x) dx[/tex]
[tex]Var(X) = E[X^2]- (E[X])^2[/tex]
Standard deviation = [tex] \sqrt[]{Var(X)}[/tex]
Mean = [tex]E[X][/tex]
Then,
[tex]E[X] = \int_{4}^{7}\frac{1}{3}dx = \frac{7^2-4^2}{2\cdot 3} = \frac{11}{2}[/tex]
[tex]E[X^2] = \int_{4}^{7}\frac{x^2}{3}dx = \frac{7^3-4^3}{3\cdot 3} = 31[/tex]
Then, [tex]Var(x) = 31-(\frac{11}{2})^2 = \frac{3}{4}[/tex]
Then the standard deviation is [tex]\frac{\sqrt[]{3}}{2}[/tex]