Probability distribution  histogram, mean, variance & standard deviation  Discrete Probabilities
Probability distribution  histogram, mean, variance & standard deviation
Lessons
Notes:
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 $= \mu2\sigma$

1.
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