Expected Value and Variance

The expected value of a random variable $X$ is the average value of $X$ over all possible outcomes.

The variance of a random variable $X$ is the expected value of the squared deviation of $X$ from its mean.

The standard deviation of $X$ is the square root of the variance of $X$, which is 4. Therefore, the standard deviation of $X$ is 2.

The probability that $X$ takes on a value between -1 and 1 is given by the area under the normal distribution curve between -1 and 1. This area is approximately 0.6827.

The probability that $X$ takes on a value greater than 12 is given by the area under the normal distribution curve to the right of 12. This area is approximately 0.1587.

The probability that $X$ takes on a value between 2 and 8 is given by the area under the normal distribution curve between 2 and 8. This area is approximately 0.9545.

The probability that $X$ takes on a value less than -2 is given by the area under the normal distribution curve to the left of -2. This area is approximately 0.0228.

The probability that $X$ takes on a value between 8 and 12 is given by the area under the normal distribution curve between 8 and 12. This area is approximately 0.6827.

The probability that $X$ takes on a value greater than 10 is given by the area under the normal distribution curve to the right of 10. This area is approximately 0.3173.

The probability that $X$ takes on a value between -3 and 3 is given by the area under the normal distribution curve between -3 and 3. This area is approximately 0.9973.

The probability that $X$ takes on a value less than 8 is given by the area under the normal distribution curve to the left of 8. This area is approximately 0.1587.

The probability that $X$ takes on a value between 0 and 10 is given by the area under the normal distribution curve between 0 and 10. This area is approximately 0.9545.

The probability that $X$ takes on a value greater than 2 is given by the area under the normal distribution curve to the right of 2. This area is approximately 0.0455.

The probability that $X$ takes on a value between 6 and 14 is given by the area under the normal distribution curve between 6 and 14. This area is approximately 0.9545.