Continuous Random Variables

Continuous Random Variables

Probability Density Function• When plotted, discrete random variables (categories)

form “bars”• A bar represents the # of times that category occurred.

Probability Density Function• As more and more different categories occur the “bars”

get thinner and thinner• If there are an infinite number of categories, the bars are

infinitesimally wide

Probability density function

Uniform distribution

Numerical Integration in R

• The integrate() function is used to numerically integrate functions in R.

Example

Numerical Integration in R

• The integrate() function is used to numerically integrate functions in R.

The cumulative distribution function

Computing probabilities using the cdf

Fig. 4-8, p. 138

F(b) F(a)

F(b) - F(a)

Example

Percentiles of a continuous distribution

Fig. 4-10, p. 139

Quantiles in RIn R, all of the built in distributions have a built in function called the quantile function which calculates percentiles. The quantile function always begins with the letter q. So for instance:

Suppose that Z has a standard normal distribution(to be introduced soon) and we wish to determine the 74th percentile of Z, i.e. the value pp such that P(Z < pp) = .74. In R we just use the qnorm() function as follows:

So P(Z<.6433454) ~ .74 To verify in R:

Mean of a continuous random variable

Expected value of a function of a rv

Variance of a continuous rv

Example

• Compute the mean of this rv• Compute the standard deviation of this rv

Example• Compute the mean of this rv

Example• Compute the standard deviation of this rv