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Continuous Random Variables. Probability Density Function When plotted, discrete random variables...

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Continuous Random Variables
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Page 1: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Continuous Random Variables

Page 2: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Continuous Random Variables

Page 3: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

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

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

Page 4: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

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

Page 5: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Probability density function

Page 6: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.
Page 7: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Uniform distribution

Page 8: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Numerical Integration in R

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

Page 9: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Example

Page 10: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.
Page 11: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Numerical Integration in R

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

Page 12: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

The cumulative distribution function

Page 13: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Computing probabilities using the cdf

Page 14: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Fig. 4-8, p. 138

F(b) F(a)

F(b) - F(a)

Page 15: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Example

Page 16: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Percentiles of a continuous distribution

Page 17: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Fig. 4-10, p. 139

Page 18: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

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:

Page 19: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Mean of a continuous random variable

Page 20: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Expected value of a function of a rv

Page 21: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Variance of a continuous rv

Page 22: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Example

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

Page 23: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Example• Compute the mean of this rv

Page 24: Continuous Random Variables. Probability Density Function When plotted, discrete random variables (categories) form “bars” A bar represents the # of.

Example• Compute the standard deviation of this rv


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