6 The Normal Distribution
Definition: Discrete Random Variable
A variable which takes on a countable number of values or finite values.
Definition: Continuous Random Variable
A variable which takes on an uncountable number of values.
Definition: Continuous Probability Distribution
A probability distribution which considers a continuous variable.
Definition: Normal Distribution
The "bell-shaped" curve.
Definition: The standard normal curve (a continuous probability distribution)
The normal distribution is a probability distribution for a continuous random variable that is "normal".
Properties of the standard normal
curve:
i. total area underneath = 1.
ii. approaches 0 asymptotically as z goes to + infinity
and - infinity.
iii. symmetric about 0.
iv. "most" area lies between -3 and 3.
Notation: za denotes the z-score having area a to its right under the standard normal curve.
Properties of a normal curve.
i. total area underneath = 1
ii. approaches 0 asymptotically as z goes to + infinity and - infinity.
iii. normal curves with parameters m and s symmetric about m.
iv. "Most" area lies between m - 3s and m + 3s .
Theorem: the normal curve with parameters m = 0 and s = 1 is the standard normal curve.
Definition: normally distributed population
If percentages for the population are (approximately) equal to the area under a normal curve (with parameters m and s).
Theorem: (Empirical Rule for a Normal Distributed Population - NDRV) - for a normally distributed population,
68.26 % of the data lies within m - s, m + s
95.44 % of the data lies within m - 2s, m + 2s
99.74 % of the data lies within m - 3s, m + 3s
Definition: normally distributed random variable
If the probabilities for the variable are (approximately) equal to area under a normal curve.
Theorem: (Empirical Rule for NDRV x)
i. P(m - s x< x < m + sx) = 0.6826
ii. P(mx - 2s x< x < mx + 2sx) = 0.9544
iii. P(m - 3s x< x < m + 3sx) = 0.9974
Theorem: (General Empirical Rule for NDRV x)
P(mx - za/2s x< x < mx + za/2sx) = 1 - a
Definition: standardized version of x
Let x be a random variable. The standardized version of x is also a random variable, z, and: z = (x - m)/sx
Theorem: NDRV with m = 0 and s = 1 has the standard normal distribution.
Theorem: If NDRV has standard normal distribution, then the probabilities are equal to the areas under a standard normal curve.
NORMAL PROBABILITY PLOTS
Theorem: For a normal probability plot with sample data:
i. if the plot is roughly linear, accept as reasonable that the population is approximately normally distributed.
ii. if the plot shows systematic deviations from linearity, conclude that the population is probably not approximately normally distributed.
Note: Outliers will fall well outside the pattern of linearity. You may want to disregard outliers, depending on several reasons.
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