Question

1. how do you find the variance and standard deviation of the probability distribution? 2. which formula is easier in finding the variance and standard deviation of the probability distribution? elaborate why.

Explanation:

Step1: Variance formula

The variance $\sigma^{2}$ of a probability distribution is given by $\sigma^{2}=\sum(x - \mu)^{2}P(x)$, where $x$ are the values of the random variable, $\mu$ is the mean ($\mu=\sum xP(x)$), and $P(x)$ is the probability of each value.

Step2: Standard deviation formula

The standard deviation $\sigma$ is the square - root of the variance, so $\sigma=\sqrt{\sum(x - \mu)^{2}P(x)}$.

For the second part, another formula for variance is $\sigma^{2}=\sum x^{2}P(x)-\mu^{2}$. This formula is often easier because it avoids calculating $(x - \mu)$ for each term. First, we calculate $\mu=\sum xP(x)$, then $\sum x^{2}P(x)$ (by squaring each $x$ value and multiplying by its probability and summing), and then subtract $\mu^{2}$.

Answer:

1. Variance: $\sigma^{2}=\sum(x - \mu)^{2}P(x)$ (where $\mu=\sum xP(x)$), Standard deviation: $\sigma=\sqrt{\sum(x - \mu)^{2}P(x)}$.
2. The formula $\sigma^{2}=\sum x^{2}P(x)-\mu^{2}$ is easier. It simplifies calculations as it avoids computing $(x - \mu)$ for each data - point. First find $\mu=\sum xP(x)$, then $\sum x^{2}P(x)$ and subtract $\mu^{2}$ to get the variance.