Stats 126 CH 7 (7.1,7.2)

probability density function (pdf)

an equation used to compute probabilities of continuous random variables. must satisfy:

1. total area under graph of the equation over all possible values must equal 1

2. the height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable

What does the area under the graph of the density function over an interval represent?

the probability of observing a value of the random variable in that interval.

model

an equation, table, or graph used to describe reality

normal curve

used to describe continuous random variables that are said to be normally distributed.

normally distributed or has a normal probability distribution

a continuous random variable is ____ if its relative frequency histogram has the shape of a normal curve.

inflection points

the points on the curve where the curvature of the graph changes

• x=mu-sigma and x=mu+sigma

properties of the normal density curve

1. it is symmetric about its mean, mu

2. because mean=median=mode, the curve has a single peak and the highest point occurs at x=mu

3. it has inflection points at mu-sigma and mu+sigma

4. the are under the curve is 1

5. the area under the curve to the right of mu equals the area under the curve to the left of mu, which equals ½

6. as x increases/ decreases without bound the graph approaches but never reaches the horizontal axis.

7. empirical rule (68% 1 std, 95% 2 std, and 99.7% 3 std)

Area under a normal curve

represents

• the proportion of the population with the characteristic described by the interval of values OR

• the probability that a randomly selected individual from the population will have the characteristic described by the interval of values

standard normal random variable

if a normal random variable X has mean different from 0 or a std different from 1, X can be transformed into a standard normal random variable Z whose mean is 0 and std is 1

Z sub alpha

the z-score such that the area under the standard normal curve to the right of “z sub alpha” is alpha