Continuous Random Variables
Recall from last week that a random variable is a variable that assigns a number to each outcome of an experiment. We only looked at discrete random variables which take on a countable number of values, with a large focus on the binomial distribution.
Now, we are looking at continuous random variables. A continuous random variable can take on an infinite number of values on an interval. Examples include weight, time to failure, or a random number on an interval. As continuous random variables can take on an uncountably infinite number of values, the probability of any one value is zero. Thus, we do not use formulas to calculate exact values, but use areas under curves known as probability density functions.
A probability density function (also known as a pdf) tells us where probabilities are concentrated and can tell us the probability of events occurring on intervals. The area under an entire pdf is equal to 1 (can you see why?), and, for random variable X, P(a < X < b) = area under the pdf from a to b. Again, the P(X = a) = 0 for any a.
Some interesting characteristics of a pdf are:
1. The mean. This is the measure of central tendency of the random variable. There is where the more common values of the random variable occur.
2. The variance. This is the measure of spread. In general, longer, "fatter", areas have more variance, than shorter narrow areas.
3, Quartiles. These are the intervals on the curve that represent the 25th, 50th, 75th percentiles. In other words, the percentile tells us the amount of probability area underneath. For example the 25th percentile is the first 25% of area under the pdf. The 50th percentile is where the half of the area is, etc. The 100th percentile would be the entire area under the curve and is technically an infinite value.
Normal Distribution
Normal
The normal distribution is easily the most important continuous probability distribution. It is known by laymen as the "bell curve", occurs naturally in many applications, and is a fundamental assumption of many statistical processes.
The normal distribution is defined by its mean, µ ("mu"), and standard deviation, No Image ("sigma"). Notice that it is symmetric about its mean and extends out to positive and negative infinity in each direction. Areas under the normal distribution are, as stated above, probabilities of X occurring on those intervals. However, areas under the curve cannot be found exactly and must be approximated using calculus.
Some common values are given in the picture above:
- About 68% of the area is within one standard deviation of the mean. In other words, there is a 68% probability that observations fall within this range.
- About 95% of the area is within two standard deviations of the mean.
- About 99.7% of the area is within three standard deviations of the mean. Thus, most of our observations should fall within this range. Values outside of this range would be uncommon, but not impossible.
Stan dard Normal Distribution
standard normal
The above percentages are also known as empirical rules and work with any normal distribution. However, if we want probabilities of different ranges we have to use approximation methods to find the area under the curve. As each µ, No Image pair would give us a different graph, it would be tedious and difficult to do this approximation every time we wanted to find probabilities. Thus, we have a nice "base" distribution, known as the standard normal distribution, to use to calculate all of our normal probabilities.
The standard normal has a mean of 0 (µ = 0) and a standard deviation of 1 ( No Image= 1). Thus, each value on the standard normal is the number of standard deviations we are from the mean, and the sign tells us if we are above or below the mean. The letter Z is reserved for any standard normal random variable. One nice feature of the standard normal is that we already know the values under the curve which are presented in tables (Z-tables are in Appendix B pp. 634-35 of your textbook). Also, modern computers can rapidly calculate areas such as here (this finds the area to the left of any Z-value).
We will look at a few examples of using the tables to find probabilities under a standard normal.
EX 1. Find the P(Z < 1.32). This is asking for the probability to the left of Z = 1.32. The tables give us the areas to the left of the Z-values, so we just need to find 1.32 on the table. The right page (635) is all of the positive Z-values so we look there. We look for 1.3 in the left column then go over to 0.02 for the last digit which gives us a value of 0.9066. Thus, the area to the left of Z = 1.32 is 0.9066 or the P(Z < 1.32) = 0.9066 or about 90.6%.
If we want areas to the right, we find the area to the left of the value and subtract it from 1 as the area under the whole curve is equal to 1.
EX 2. Find the P(Z > 0.93). This is asking for the area to the right of Z = 0.93. First, we find the area to the left of 0.93. From the table we get 0.8238. Then we find 1-0.8238 = 0.1662. Thus, the area to the right of Z = 0.93 is 0.1662 or the P(Z > 0.93) = 0.1662.
Finally, if we want values between two Z-values, we find the area to the left of each one and subtract the smaller value from the larger value.
EX 3. Find the P( -1.21 < Z < 0.73). This is asking for the area between Z = -1.21 and Z = 0.73. We find both of the areas to the left directly from the table: the area to the left of -1.21 is 0.1131 and the area to the left of 0.73 is 0.7673. Thus, the area between the two points is 0.7673-0.1131 = 0.6542. So the P(-1.21 < Z < 0.73) = 0.6542.
Standardizing
The Z-tables only find values for standard normal variables. However, what if we want to find the area under any normal curve for X for any mean and variance? What we do is transform X into a standard normal variable and find the areas for Z-values. Thus, we can find the area under any normal curve for any values since we only need one set of tables.
The transformation formula for any normal X-value is :
No Image
In English, what we do is subtract X's mean value from our X value on top. What this does is find our distance from the mean. Then we divide by the standard deviation; this gives us the number of standard deviations away our X value is from the mean. Which is also known as a standard normal variable, Z! Now we can calculate any X probabilities.
EX 4. We are given that X is a normal variable with µ = 25 and No Image= 10. Find the P(12.5 < X < 40).
First we need to transform the X values into Z values.
No Image
Now, using the z-tables we get: P(-1.25 < Z < 1.5) = P(Z < 1.5) - P(Z < -1.25) = 0.9332 - 0.1056 = 0.8276. So the probability X is between 12.5 and 40 is about 82.8%.
Distributions of Sample Means and the Central Limit Theorem
Now we are going to focus on the distribution of a sample mean. Consider taking a random sample of size n from a population and all the observations, No Image , are independent of each other. A sample mean, No Image ("X bar"), can be calculated as such:
No Image
Nothing surprising here as we know that to find a mean you add all of the observations and divide by the number of them. However, say we kept taking different random samples and calculating new means. It should be apparent that, generally, each mean we calculate will be different from the others as we will have different observations in each sample. Thus, the sample means, themselves, have a probability distribution.
How can we find their distribution? For example, we may want to find the probability that our next sample mean exceeds 100. Fortunately, we have a powerful and famous theorem, the Central Limit Theorem (CLT) that answers this question for us. (The proof of the CLT is beyond the scope of this course, but may be found online.)
Briefly, the CLT states that if we take a "large enough" random sample, n, from a population with mean, µ, and standard deviation, No Image , then the sample mean, No Image , as calculated above, is distributed as a normal variable with mean µ and standard deviation No Image . So, in practice, we can use a normal distribution to find probabilities of sample means.
Some comments. "Large enough" is a fuzzy idea, but generally, n > 30 is the rule of thumb. In this class, we can assume n is large enough for the problems. The strength of the CLT is that we do not need to know how the population is distributed; it could be anything. However, we do need to know its mean and variance, or assume values for them if not known.
Theoretical concerns aside, what you need to know is how to calculate probabilities of sample means. Let's do an example.
EX 5. A population has a mean, µ = 100 and standard deviation, No Image= 25. We take a random sample of n = 25 and calculate a sample mean, No Image. What is the probability our sample mean exceeds 110?
We want the P(No Image> 110). Without the CLT, we wouldn't know where to start. However, by the CLT, we can say that No Image is a normal random variable with mean, µ = 100 and standard deviation of
No Image = 25/5 = 5. So now we can find the probability by standardizing to Z and using the tables as we have been doing.
Thus, P( No Image> 110) = P(Z > 2) (do you see why?).
Using the Z-table we get P(Z > 2) = 1 - P(Z < 2) = 1- 0.9772 = 0.0228 or about 2.3%. So not very likely but not unheard of.
vroins
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