1. Let's start by stating the problem: We want to find the moments of a random variable $X$. 2. The $n$-th moment of a random variable $X$ is defined as $E[X^n]$, which is the expected value of $X$ raised to the power $n$. 3. The formula for the $n$-th moment is: $$\mu_n = E[X^n] = \int_{-\infty}^{\infty} x^n f_X(x) \, dx$$ where $f_X(x)$ is the probability density function (pdf) of $X$. 4. Important rules: - Moments provide information about the shape of the distribution. - The first moment ($n=1$) is the mean. - The second central moment (variance) is $E[(X - E[X])^2]$. 5. To compute moments, you need the pdf or pmf of $X$. Then, raise $X$ to the power $n$, multiply by the pdf, and integrate (or sum) over all possible values. 6. Example: If $X$ is a continuous random variable with pdf $f_X(x)$, then the first moment is: $$\mu_1 = E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx$$ 7. Similarly, the second moment is: $$\mu_2 = E[X^2] = \int_{-\infty}^{\infty} x^2 f_X(x) \, dx$$ 8. If you want central moments, subtract the mean before raising to the power: $$\mu_n' = E[(X - \mu_1)^n]$$ 9. Without a specific distribution or data, this is the general method to find moments. Final answer: Moments are calculated using the formula $$\mu_n = E[X^n] = \int x^n f_X(x) \, dx$$ for continuous variables or $$\mu_n = \sum x^n P(X=x)$$ for discrete variables.