As the title of this page suggests, we will now focus on using the normal distribution to approximate binomial probabilities. The Central Limit Theorem is the tool that allows us to do so. As usual, we'll use an example to motivate the material.
There is really nothing new here. Doing so, we get:. That is, there is a Note, however, that Y in the above example is defined as a sum of independent, binary approximation distributed random variables.
Binary approximation, as long as n is sufficiently large, we can use the Central Limit Theorem to calculate probabilities for Y. Specifically, the Central Limit Theorem tells us that:. Let's use the normal distribution then to approximate some probabilities for Y. First, binary approximation in our case that the mean is:. Such an adjustment is called a " continuity correction. Let's try a few more binary approximation. Now again, once we've made the continuity correction, the calculation reduces to a normal probability calculation:.
By the way, you might find it interesting to note that the approximate normal probability is quite close to the exact binomial probability.
We showed that the approximate probability is 0. Let's try one more approximation. Again, once we've made the continuity correction, the calculation reduces to a normal probability calculation:.
By the way, the exact binomial probability is 0. Just a couple of comments before we close our discussion of the normal approximation to the binomial.
The general rule of thumb is that the sample size n is "sufficiently large" if: Because our sample size was at least 10 well, barely! Then, the two conditions are met if:. Binary approximation that mean all of our discussion here is for naught? No, not at all! In reality, we'll binary approximation often use the Central Limit Theorem as applied to the sum of independent Bernoulli random variables to help us draw conclusions about binary approximation true population proportion p.
If binary approximation take the Z random variable that we've been dealing with above, and divide the numerator by n and the denominator by n and thereby not changing the overall quantitywe get the following result: You'll definitely be seeing much more of this in Stat !
Eberly College binary approximation Science. Approximations binary approximation Discrete Binary approximation. Printer-friendly version As the title of this page suggests, we will now focus binary approximation using the normal distribution to approximate binomial probabilities.
Doing so, we get: First, recognize in our case that the mean is: Introduction to Probability Section 2: Discrete Distributions Binary approximation 3: Continuous Distributions Section 4: Bivariate Distributions Section 5: Distributions of Functions of Random Variables Lesson Functions of One Random Variable Lesson Transformations of Two Random Variables Lesson Several Independent Random Variables Lesson The Central Limit Theorem Lesson Hypothesis Testing Section 8: Nonparametric Methods Section 9: Bayesian Methods Section
The binomial approximation is useful for approximately calculating powers of sums of a small number and 1. This can greatly simplify mathematical expressions see example and is a common tool in physics. This approximation can be proven several ways including the binomial theorem and ignoring the terms beyond the first two. Thus, standard linear approximation tools from calculus apply: This result from the binomial approximation can always be improved by keeping additional terms from the Taylor Series above.
While the expression is small, it is not exactly zero. It is possible to extract a nonzero approximate solution by keeping the quadratic term in the Taylor Series, i.
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