Two Common Distributions Help (page 2)
Introduction to Two Common Distributions
The nature of a probability density function can be described in the form of a distribution. Let's get a little bit acquainted with two classical types: the uniform distribution and the normal distribution. These are mere introductions, intended to let you get to know what sorts of animals probability distributions are. There are many types of distributions besides these two. Distributions are sort of like spiders. In one sense, ''when you've seen one, you've seen them all.'' But if you are willing to go deep, you can look at any species endlessly and keep discovering new things about it.
In a uniform distribution, the value of the function is constant. When graphed, it looks ''flat,'' like a horizontal line (Fig. 3-9).
Let x be a continuous random variable. Let xmin and xmax be the minimum and maximum values that x can attain, respectively. In a uniform distribution, x has a density function of the form:
- f(x) = 1 / (xmax – xmin)
Because the total area under the curve is equal to 1, the probability Pab that any randomly chosen x will be between a and b is:
- Pab = (b – a) / (xmax – xmin)
Suppose that the experiment described above reveals that equal numbers of people always have each of the given tested blood-pressure numbers between two limiting values, say xmin = 100 and xmax = 140. Imagine that this is true no matter how many people are tested, and no matter how many different values of blood pressure are specified within the range xmin = 100 and xmax = 140. This is far-fetched and can't represent the real world, but if you play along with the idea, you can see that such a state of affairs produces a uniform probability distribution.
The mean (μ), the variance (σ2), and the standard deviation (σ), which we looked at in the last chapter for discrete random variables, can all be defined for a uniform distribution having a continuous random variable. A detailed analysis of this is beyond the scope of this introductory course. But here are the formulas, in case you're interested:
- μ = (a + b) / 2
- σ2 = (b – a)2 / 12
- σ = [(b – a)2 / 12]1/2
In a normal distribution, the value of the function has a single central peak, and tapers off on either side in a symmetrical fashion. Because of its shape, a graph of this function is often called a bell-shaped curve (Fig. 3-10). The normal distribution isn't just any bell-shaped curve, however. In order to be a true normal distribution, the curve must conform to specific rules concerning its standard deviation.
The symbol σ represents the standard deviation of the function, which is an expression of the extent to which the values of the function are concentrated. It's the same concept you saw in Chapter 2, but generalized for continuous random variables. A small value of σ produces a ''sharp'' curve with a narrow peak and steep sides. A large value of σ produces a ''broad'' curve with less steep sides. As σ approaches 0, the curve becomes narrower and narrower, closing in on a vertical line. If σ becomes arbitrarily large, the curve becomes almost flat and settles down near the horizontal axis. In any normal distribution, the area under the curve is equal to 1, no matter how much or little it is concentrated around the mean.
The symbol μ represents the mean, or average. Again, this is the same mean you learned about in Chapter 2, but generalized for continuous random variables. The value of μ can be found by imagining a moving vertical line that intersects the x axis. When the position of the vertical line is such that the area under the curve to its left is 1/2 (or 50%) and the area under the curve to its right is also 1/2 (50%), then the vertical line intersects the x axis at the point x = μ. In the normal distribution, x = μ at the same point where the function attains its peak, or maximum, value.
The Empirical Rule
Imagine two movable vertical lines, one on either side of the vertical line x = μ. Suppose these vertical lines, x = a and x = b, are such that the one on the left is the same distance from x = μ as the one on the right. The proportion of data points in the part of the distribution a < x < b is defined by the proportion of the area under the curve between the two movable lines x = a and x = b. Figure 3-10 illustrates this situation. A well-known theorem in statistics, called the empirical rule, states that all normal distributions have the following three characteristics:
- Approximately 68% of the data points are within the range ±σ of μ.
- Approximately 95% of the data points are within the range ±2σ of μ.
- Approximately 99.7% of the data points are within the range ±3σ of μ.
Two Common Distributions Practice Problems
Suppose you want it to rain so your garden will grow. It's a gloomy morning. The weather forecasters, who are a little bit weird in your town, expect a 50% chance that you'll see up to 1 centimeter (1 cm) of rain in the next 24 hours, and a 50% chance that more than 1 cm of rain will fall. They say it is impossible for more than 2 cm to fall (a dubious notion at best), and it is also impossible for less than 0 cm to fall (an absolute certainty!). Suppose the radio disc jockeys (DJs), who are even weirder than the meteorologists, announce the forecast and start talking about a distribution function called R(x) for the rain as predicted by the weather experts. One DJ says that the amount of rain represents a continuous random variable x, and the distribution function R(x) for the precipitation scenario is a normal distribution whose value tails off to 0 at precipitation levels of 0 cm and 2 cm. Draw a crude graph of what they're talking about.
See Fig. 3-11. The portion of the curve to the left of the vertical line, which represents the mean, has an area of 0.5. The mean itself is x = σ = 1 cm.
Fig. 3-11. Illustration for Practice 1.
Imagine the scenario above, as the DJs continue to expound. They start talking about the extent to which the distribution function is spread out around the mean value of 1 cm. One of them mentions that there is something called standard deviation, symbolized by a lowercase Greek letter called sigma that looks like a numeral 6 that has fallen over. Another DJ says that 68% of the total prospective future wetness of the town falls within two values of precipitation defined by sigma on either side of the mean. Draw a crude graph of what they're talking about now.
See Fig. 3-12. The shaded region represents the area under the curve between the two vertical lines represented by x = μ – σ and x = μ + σ. This is 68% of the total area under the curve, centered at the vertical line representing the mean, x = μ.
Fig. 3-12. Illustration for Practice 2 and 3.
What does the standard deviation appear to be, based on the graph of Fig. 3-12?
It looks like approximately σ = 0.45, representing the distance of either the vertical line x = μ – σ or the vertical line x = μ + σ from the mean, x = μ = 1 cm. Note that this is only the result of the crude graph we've drawn here. The DJs have not said anything about the actual value of σ. We want to stick around and find out if they'll tell us what it is, but then there is a massive lightning strike on the broadcast tower, and the station is knocked off the air.
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