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4. PROBABILITY DISTRIBUTIONS
47
Thus f (x) meets the requirements of being a probability density function.
Finally, for every real number a,
Z a
P (X = a) =
f (x) dx = 0.
a
We have been using normal curves. Let’s take a closer look at them.
Definition. A normal curve with mean µ and standard deviation is the graph of the function
(x µ)2
1
f (x) = p e 2 2
2⇡
Thus normal curves are completely determined by their mean and standard
deviation.
For the three normal curves above, one has a mean of 70 and a standard
deviation of 5, another has a mean of 70 and a standard deviation of 10, and
the third has a mean of 50 and a standard deviation of 10. Which is which?
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