In this video I go over an introduction into probability and how we can model probability through looking at the proportion of specific ranges of any continuous random variable over many experimental trial tests. From the experimental data, I show how we can create a model of the data, called the probability density function. This function is unique in that the area under the curve represents the probability that the random variable lies in that interval. Since we are dealing with areas under a curve, this naturally ties well into integrals. This is a very interesting and in-depth video that goes over the very core and basis of probability so make sure to watch this video!
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Applications of Integrals: Probability
Calculus plays a role in the analysis of random behavior.
Suppose we consider the cholesterol level of a person chosen at random from a certain age group, or the height of an adult female chosen at random, or the lifetime of a randomly chosen battery of a certain type.
Such quantities are called continuous random variables because their values actually range over an interval of real numbers, although they might be measured or recorded only to the nearest integer.
- A continuous random variable is a random variable where the data can take infinitely many values.
- It is also impossible to measure exactly any of the values but can estimate it based on how accurate your measurement device is.
- The opposite of this is something like a coin flip, where it's either heads or tails and no in-between.
We might want to know the probability that a blood cholesterol level is greater than 250, or the probability that the height of an adult female is between 60 inches and 70 inches, or the probability that the battery we are buying lasts between 100 and 200 hours.
If X represents the lifetime of that type of battery, we denote this last probability as follows:
We can interpret this probability as the long-run proportion of all batteries of a specified type whose lifetimes are between 100 and 200 hours.
In other words, if you were to run tests on many batteries, we will use the frequency or number of batteries with lifetimes in any given range as the basis for the probability of a battery lifetime in that range
For example:
-- Out of 100,000 battery tests, 80,000 batteries have a lifetime between 100 and 200 hours.
-- Since 100,000 can be assumed to be adequately large, we can use the proportion of 80,000 batteries having lifetimes between 100 and 200 hours to the 100,000 total batteries as the probability that any given battery will have a lifetime between 100 to 200 hours..
-- Since it represents a proportion, the probability naturally falls between 0 and 1.
-- In this case:
Notice in the above example, that we used a range of lifetimes.
If rather we were to find the probability that a battery has exactly 150 hours lifetime, the probability is actually 0!
This is because there is no way that a battery has a lifetime of exactly 150.0000000000000…. hours.
Thus instead of finding the probability of exactly 150 hours, we instead could look at approximately 150 hours by instead finding the probability that a battery has a lifetime between 150 to 150.01 hours. Let's assume that the answer is 2% (or in our case 2,000 batteries out of 100,000 total batteries).
Now let's say we want to find the probability that a battery dies between 150 and 150.001 hours? And between 150 and 150.0001 hours?
In these cases since we are considering such small intervals spaced out at 1/10th the previous interval, we can assume that the probabilities are also going to be approximately 1/10th the previous probability. That is:
In this simple case, the ratio of (probability of a battery dying during an interval) / (duration of that interval) is approximately constant:
Thus if the interval, Δx, is small, we can assume that the above ratio is constant and in general we have:
We can rearrange this above formula to get the probability of a battery with an approximate lifetime of 150 hours:
For example, if we wanted to find the probability that a battery dies at 150 hours but accurate to 1 nanosecond (1 hour = 3.6x1012 ns), the probability is roughly:
The ratio of 2 hour-1 is called the probability density at a battery lifetime of 150 hours.
If we were to graph the probability density of all possible battery lifetimes, we will obtain the probability density function.
- Remember that these probability densities are taken from the frequency interpretation (or ratio of specific battery lifetimes to total batteries) as explained above
- This function is numerically modeled or derived from experimentally measured values.
- Most familiar probability density function is known as the Bell Curve and models many continuous random variables well, such as population heights, grades, etc.
- Let's assume that the bell curve models our battery lifetimes random variable well.
As can be seen above, the probability that a battery has a lifetime of approximately 150 hours is represented as the area of the rectangle under the curve width Δx and height 2/hour.
Now let's look at a more general probability density function to see if we can approximate the probability that the continuous random variable, x, lies between a and b.
We can approximate this by breaking up the interval into n subintervals of equal width Δx and select the sample point, xi*, as the left endpoint, xi-1.
Since the probability that x is between xi-1 and xi is approximated by the area Ai we can thus sum all these areas to get the overall probability that x lies between a and b.
Taking the limit and writing this as an integral, we obtain:
In general, the probability density function f of a random variable X satisfies the following conditions:
- f(x) ≥ 0 for all x.
- Because probabilities are measured on a scale from 0 to 1, and that the area under the curve represents a probability for any given range, the probability that the random variable, X, is any value has to be 100%, and this represents the entire area under the curve. This can be written as an integral:
