Chapter 7 Lesson 7.3 Random Variables and Probability Distributions 7.3 Probability Distributions for Continuous Random Variables

Probability Distributions for Continuous Random Variables

Consider the random variable: x = the weight (in pounds) of a full-term newborn child Suppose that weight is reported to the nearest pound. The following probability histogram displays the distribution of weights. Now suppose that weight is reported to the nearest 0.1 pound. This would be the probability histogram. What type of variable is this? Notice that the rectangles are narrower and the histogram begins to have a smoother appearance. If weight is measured with greater and greater accuracy, the histogram approaches a smooth curve. This is an example of a density curve.

Probability Distributions for Continuous Random Variable Is specified by a curve called a density curve. The probability of observing a value in a particular interval is the area under the curve and above the given interval. The total area under the density curve equals one.

Let x denote the amount of gravel sold (in tons) during a randomly selected week at a particular sales facility. Suppose that the density curve has a height f(x) above the value x, where The density curve is shown in the figure:

.75 Gravel problem continued... What is the probability that at most ½ ton of gravel is sold during a randomly selected week? P(x < ½) = The probability would be the shaded area under the curve and above the interval from 0 to 0.5.

=.75 Gravel problem continued... What is the probability that less than ½ ton of gravel is sold during a randomly selected week? P(x < ½) = Does the probability change whether the ½ is included or not? P(x < ½) Hmmm... This is different than discrete probability distributions where it does change the probability whether a value is included or not!

Suppose x is a continuous random variable defined as the amount of time (in minutes) taken by a clerk to process a certain type of application form. Suppose x has a probability distribution with density function: The following is the graph of f(x), the density curve: Time (in minutes) Density

Application Problem Continued... What is the probability that it takes more than 5.5 minutes to process the application form? Time (in minutes) Density P(x > 5.5) =.5(.5) =.25 Find the probability by calculating the area of the shaded region (base × height). When the density is constant over an interval (resulting in a horizontal density curve), the probability distribution is called a uniform distribution.

Other Density Curves Some density curves resemble the one below. Integral calculus is used to find the area under the these curves. Don’t worry – we will use tables (with the values already calculated). We can also use calculators or statistical software to find the area.

The probability that a continuous random variable x lies between a lower limit a and an upper limit b is P(a < x < b) = (cumulative area to the left of b) – (cumulative area to the left of a) P(a < x < b) = P(x < b) – P(x < a) This will be useful later in this chapter!

Homework Pg.414: #