A probability density function of a random variable is given by $f(x)$ below. Find the mean, variance, and standard deviation. \[ f(x)=\left\{\begin{array}{ll} \frac{9}{x^{10}} & \text { if } x \geq 1 \\ 0 & \text { otherwise } \end{array}\right. \] The variance is $V(X)=$ (Type an integer or decimal rounded to two decimal places as needed.)

Step 1 ：The probability density function of a random variable is given by \(f(x)=\frac{9}{x^{10}}\) for \(x \geq 1\) and 0 otherwise.

Step 2 ：The variance of a random variable X is given by the formula \(V(X) = E(X^2) - [E(X)]^2\), where \(E(X)\) is the expected value or mean of X, and \(E(X^2)\) is the expected value of X squared.

Step 3 ：To find \(E(X)\) and \(E(X^2)\), we need to integrate \(x*f(x)\) and \(x^2*f(x)\) over the range of x, respectively. In this case, the range of x is from 1 to infinity.

Step 4 ：By performing the integrations, we find that \(E(X) = \frac{9}{8}\) and \(E(X^2) = \frac{9}{7}\).

Step 5 ：Substituting these values into the formula for variance, we find that \(V(X) = \frac{9}{7} - \left(\frac{9}{8}\right)^2 = \frac{9}{448}\).

Step 6 ：Final Answer: The variance of the random variable X is given by \(\boxed{\frac{9}{448}}\).