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Normal Distributions
So far we have dealt with random variables with a finite number of possible values. For example; if X is the number of heads that will appear, when you flip a coin 5 times, X can only take the values 0, 1 , 2 , 3 , 4 , or 5.
Some variables can take a continuous range of values, for example a variable such as the height of 2 year old children in the U.S. population or the lifetime of an electronic component. For a continuous random variable X, the analogue of a histogram is a continuous curve (the probability density function) and it is our primary tool in finding probabilities related to the variable. As with the histogram for a random variable with a finite number of values, the total area under the curve equals 1.
Normal Distributions
Probabilities correspond to areas under the curve and are calculated over intervals rather than for specific values of the random variable. Although many types of probability density functions commonly occur, we will restrict our attention to random variables with Normal Distributions and the probabilities will correspond to areas under a Normal Curve (or normal density function).
This is the most important example of a continuous random variable, because of something called the Central Limit Theorem: given any random variable with any distribution, the average (over many observations) of that variable will (essentially) have a normal distribution. This makes it possible, for example, to draw reliable information from opinion polls.
Properties of a Normal Curve
- All Normal Curves have the same general bell shape.
- The curve is symmetric with respect to a vertical line that passes through the peak of the curve.
- The curve is centered at the mean μ which coincides with the median and the mode and is located at the point beneath the peak of the curve.
- The area under the curve is always 1.
- The curve is completely determined by the mean μ and the standard deviation σ. For the same mean, μ, a smaller value of σ gives a taller and narrower curve, whereas a larger value of σ gives a flatter curve.
- The area under the curve to the right of the mean is 0.5 and the area under the curve to the left of the mean is 0.5.
Properties of a Normal Curve
- The empirical rule (68%, 95%, 99.7%) for mound shaped data applies to variables with normal distributions. For example, approximately 95% of the measurements will fall within 2 standard deviations of the mean, i.e. within the interval (μ − 2 σ, μ + 2σ).
- If a random variable X associated to an experiment has a normal probability distribution, the probability that the value of X derived from a single trial of the experiment is between two given values x 1 and x 2 (P(x 1 6 X 6 x 2 )) is the area under the associated normal curve between x 1 and x 2. For any given value x 1 , P(X = x 1 ) = 0, so P(x 1 6 X 6 x 2 ) = P(x 1 < X < x 2 ).
The standard Normal curve
The standard Normal curve is the normal curve with mean μ = 0 and standard deviation σ = 1.
We will see later how probabilities for any normal curve can be recast as probabilities for the standard normal curve. For the standard normal, probabilities are computed either by means of a computer/calculator of via a table.
Areas under the Standard Normal Curve
z
Area = A(z) = P(Z 6 z)
Probabilities for the standard Normal
The shaded area is A(1) = 0.8413, correct to 4 decimal places.
The section of the table shown above tells us that the area under the standard normal curve to the left of the value z = 1 is 0.8413. It also tells us that if Z is normally distributed with mean μ = 0 and standard deviation σ = 1, then P(Z 6 1) = .8413.
Examples
If Z is a standard normal random variable, what is P(Z 6 2)? Sketch the region under the standard normal curve whose area is equal to P(Z 6 2). Use the table to find P(Z 6 2).
P(Z 6 2) = 0.9772.
Area to the right of a value
Recall now that the total area under the standard normal curve is equal to 1. Therefore the area under the curve to the right of a given value z is 1 − A(z). By the complement rule, this is also equal to P(Z > z).
z
Area = 1 − A(z) = P(Z > z)
Examples
If Z is a standard normal random variable, use the above principle to find P(Z > 2). Sketch the region under the standard normal curve whose area is equal to P(Z > 2). P(Z 6 2) = 0.9772 so P(Z > 2) = 1 − 0 .9772 = 0.0228.
The area between two values
We can also use the table to compute P(z 1 < Z < z 2 ) = P(z 1 6 Z < z 2 ) = P(z 1 < Z 6 z 2 ) = P(z 1 6 Z 6 z 2 ) = A(z 2 ) − A(z 1 ).
z 1 z 2
Area = A(z 2 ) − A(z 1 ) = P(z 1 < Z < z 2 )
Our previous examples can be thought of like this: P(Z 6 z) = P(−∞ < Z 6 z) = A(z) − A(−∞) = A(z) P(z < Z) = P(z < Z < ∞) = A(∞) − A(z) = 1 − A(z)
Example
If Z is a standard normal random variable, find P(− 3 6 Z 6 3). Sketch the region under the standard normal curve whose area is equal to P(− 3 6 Z 6 3).
P(− 3 6 Z 6 3) = P(Z 6 3) − P(Z 6 −3) =
- 9987 − 0 .0013 = 0.9973.
Verifying the empirical rule
P(− 1 6 Z 6 1) = P(Z 6
1)−P(Z 6 −1) = 0. 8413 −
P(− 2 6 Z 6 2) = P(Z 6
2)−P(Z 6 −2) = 0. 9772 −
Examples
(a) Sketch the area beneath the density function of the standard normal random variable, corresponding to P(− 1. 53 6 Z 6 2 .16), and find the area. P(− 1. 53 6 Z 6 2 .16) = P(Z 6 2 .16) − P(Z 6 − 1 .53) =
- 9846 − 0 .0630 = 0.9216.
(b) Sketch the area beneath the density function of the standard normal random variable, corresponding to P(−∞ 6 Z 6 1 .23) and find the area.
P(−∞ 6 Z 6 1 .23) = 0.8907.
(c) Sketch the area beneath the density function of the standard normal random variable, corresponding to P(1. 12 6 Z 6 ∞) and find the area. P(1. 12 6 Z 6 ∞) = 1 −
P(Z 6 1 .12)
