Probability distribution - histogram, mean, variance & standard deviation

Probability distribution - histogram, mean, variance & standard deviation

Lessons

For a probability distribution:
\cdot mean:μ=[xp(x)]mean:\mu = \sum [x \cdot p(x)]
\cdot variance:σ2=[(xμ)2p(x)]=[x2p(x)]μ2variance:\sigma^2 = \sum [(x-\mu)^2 \cdot p(x)]= \sum[x^2 \cdot p(x)] - \mu^2
\cdot standarddeviation:σ=σ2=[(xμ)2p(x)]=[(x2p(x)]μ2standard\;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 =μ+2σ= \mu+2\sigma
\cdot minimum usual value =μ2σ= \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”.