Continuous Random Variables

Section 7.1.2

Starter 7.1.2

• A 9-sided die has three faces that show 1, two faces that show 2, and one face each showing 3, 4, 5, 6.

• Let X be the number that shows face-up.

• Draw the PDF histogram of X.

Objectives• Display the PDF of a continuous random

variable as a density curve.• Find the probability that a continuous

random variable is within a specified interval by finding the area under a density curve.

California Standard 4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

Definitions

• A random variable is a variable whose value is the numerical outcome of a random event.

• A continuous random variable can take on ANY value within a specified interval.

• The probability distribution function (PDF) of a continuous r.v. is represented by a density curve.

The Probability of a Continuous R.V.

• Consider the spinner on the overhead projector.

• When I spin it, what is the probability that it lands on a number between .3 and .5?

• What is the probability that it lands on a number between 0 and 1?

• Draw a density curve that shows the possible outcomes and probabilities.– You are drawing the PDF of X

The Density Curve of a Continuous R.V.

• For the spinner, you drew a rectangle whose X values go from 0 to 1 and whose Y values are always exactly 1.– Recall that this is called a uniform distribution

• The probability of an event is the area under the curve and above the range of X values that make up the event.– So P(.3 x .5) is 0.2 because that is the

area under the curve above that range of x

The Probability of One Exact Outcome• When I spin the spinner, what is the probability

of hitting 0.3 exactly?– In notation: evaluate P(X = 3)

• No matter how close I come, a really precise measurement will show some error.

• Example: How tall are you EXACTLY?• Example: Watch the TI try to match an exact

number with its random number generator.– Run program CONTRV

• Conclusion: The probability of any exact outcome is always zero!– Note that the area above a point (like x = .3) is zero– So P(a x b) is the same as P(a < x < b)

A New Problem

• For a certain random variable the density curve that follows the line y = x– Draw the PDF of X (i.e. the density curve)

• You should have drawn a triangle that goes from the origin to (√2, √2)

• Find P(0 < x < 1)– .5(1)(1) = .5

Another Problem• Suppose we know the heights of Northgate

girls to be N(166, 7.3) cm– Remember what that means?

• Find P(height < 155)• A normal distribution is a density curve, so

draw a normal curve and find the area below h = 155– Normalcdf(0, 155, 166, 7.3) = .0659– So P(h < 155) = 6.6%– We could also find z = (155 – 166) / 7.3 = -1.51

and use Table A

Objectives• Display the PDF of a continuous random

variable as a density curve.• Find the probability that a continuous

random variable is within a specified interval by finding the area under a density curve.

California Standard 4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

Homework

• Read pages 375 – 378

• Do problems 4 and 5