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Lecture 11

# Lecture 11.docx

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McMaster University

Economics

ECON 2B03

Jeff Racine

Fall

Description

Lecture 11
Continuous Probability Density Functions
Continuous random variables
Unlike their discrete counterparts, continuous random variables are ‘uncountable’
and can assume any value on the real number line
Rather than looking at p(X =x) (i.e. the probability of any specific occurrence), we
instead look at p( a < X < b), i.e., the probability that a continuous random variable lies in
an interval
The probability density function
Commonly shows probabilities with ranges of values along the continuum of
possible values that the random variables might take on
Let some smooth curve represent how the probability of a continuous random
variable X is distributed. If the smooth curve can be represented by a formula f(X), then
the function f(X) is called the probability density function
Areas will be seen to play an important role in determining probabilities
Since there exist an infinite number of points in an interval, we cannot assign a non-zero
probability to each point and still have their probability sum to one, therefore, the
probability of any specific occurrence of a continuous random variable is defined to be
zero
The total area under the probability density function (just as under the relative frequency
histogram) must therefore equal 1
We shall study two common distributions
The uniform probability distribution
The normal probability distribution
Basic Rules for Probability Density Functions
If a smooth curve is to represent a probability density function, then the following two
requirements must be met:
The total area between the curve and the horizontal axis must be unity, i.e.,
∞
∫ f (x)dx=1
−∞ The curve must never fail below the horizontal axis, i.e., f(x)

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