# Probability distribution - histogram, mean, variance & standard deviation

### Probability distribution - histogram, mean, variance & standard deviation

#### Lessons

For a probability distribution:
$\cdot$ $mean:\mu = \sum [x \cdot p(x)]$
$\cdot$ $variance:\sigma^2 = \sum [(x-\mu)^2 \cdot p(x)]= \sum[x^2 \cdot p(x)] - \mu^2$
$\cdot$ $standard\;deviation: \sigma = \sqrt{\sigma^2}= \sqrt{\sum [(x-\mu)^2 \cdot p(x)]} = \sqrt{\sum [(x^2 \cdot p(x)]- \mu^2}$

Range Rule of Thumb (Usual VS. Unusual):
$\cdot$ maximum usual value $= \mu+2\sigma$
$\cdot$ minimum usual value $= \mu-2\sigma$
• 1.
Discrete VS. Continuous

• 2.
Probability Histogram, Mean, Variance and Standard Deviation
The following table gives the probability distribution of a loaded (weighted) die:
 outcome probability 1 0.05 2 0.10 3 0.30 4 0.33 5 0.15 6 0.07
a)
Create a probability distribution histogram.

b)
Using statistics formulas to find the mean, variance, and standard deviation of the probability distribution.

c)
Using calculator commands to find the mean, variance, and standard deviation of the probability distribution.

d)
Based on the “range rule of thumb”, determine the outcomes that are considered as “usual” and “unusual”.