the graph of a continuous probability distribution is a curve with the
property that the probability of a continuous random variable \( X \) can be
represented by the area under the curve
features of a continuous probability distribution
the outcomes are measured, not counted
the entire area under the curve and above the \( x \)-axis is \( 1 \)
probability is calculated by intervals of \( x \), not values
\( P(c < x < d) \) represents the probability that the value of a random
variable \( X \) is in the interval between the values \( c \) and \( d \),
\( P(c < x < d) \) represents the area under the curve \( f(x) \), above
the \( x \)-axis, between \( c \) and \( d \)
\( P(x=c) \) is always \( 0 \) and \( P(x < c) \) is the same as
\( P(x \le c) \)
the distribution is defined by a probability density function (pdf)
\( f(x) \) that defines the graph of the curve
the area under the curve is given by a cumulative distribution function (cdf)
there are many continuous probability functions that can be studied, the simplest
example is a uniform distribution
\( X\mbox{~}U(a,b) \), "\( X \) is a random variable with a uniform
probability distribution from \( x=a \) to \( x=a \)"
in a uniform probability distribution all events are equally likely
the probability distribution function for the uniform distribution
\( X\mbox{~}U(a,b) \) is \( \displaystyle{f(x)=\frac{1}{b-a}} \),
\( a \le x \le b \)
the mean and standard deviation for a uniform probability distribution
\( X\mbox{~}U(a,b) \)
are \( \displaystyle{\mu = \frac{1}{2}(a+b)} \) and \( \sigma = \sqrt{\frac{(b-a)^{2}}{12}} \)
later we will concentrate primarily on normal distributions
Exam 1
Exam 1 is taken in class on Wednesday, February 15.