What are some uses of the binomial distribution
This article covers the topic Binomial distribution. Here you get one first definition the Binomial distribution. Then we explain them Formulas the distribution and will calculate various tasks using a few examples.
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Binomial distribution explained simply
What is a Binomial distribution? Like the syllable "Bi"(Lat. Two) already suggests, everything here revolves around a pair of terms, namely" yes or no ". Did I get a hit or not? Do I have a success or a failure? Such “either or” experiments with only 2 possible results are based on the binomial distribution. This is also calledBernoulli experiments.
A classic example of such an experiment would be a coin toss where you can only get heads or tails. With the help of Probability function and the Distribution functionthe Binomial distribution can you Bernoulli experiments and determine, for example, how likely it is that you will land k hits with n throws.
Binomial Distribution Definition
The Binomial distribution is one of the most important discrete distributions. A binomially distributed random experiment is created by repeating a Bernoulli experiment n times. So one only differentiates between success and failure.
Occasionally the Binomial distribution also as Binomial distribution designated. This designation is self-evident not correct!
Binomial distribution formula
You can adjust the density with the following Probability function describe:
When X is a binomial random variable , then
as the Probability function the Binomial distribution Are defined. The parameter n stands for the number of draws, p for the probability of a success or hit and k for the number of successes. Note that k is often abbreviated with lowercase x, depending on the author. So don't let the name confuse you.
The following can also be used as an alternative notation:
As you can see, the different notation does not change anything in the actual calculation.
As already mentioned, the parameter k represents the number of successes or hits (depending on the context). The expression stands for Binomial coefficients. This is also in the Combinatorics used. You can calculate it with the following formula:
The Probability function can of course also be displayed graphically. Here you can see a random experiment with 5 draws and a probability of success of 0.5.
The probability function described above is only defined for non-negative k-values. The negative binomial distribution is a special case with main application in actuarial mathematics.
Cumulative binomial distribution
To the Distribution function To calculate the probabilities, you can either add up the probabilities manually or, if available, from one Binomial distribution table (also called distribution table).
In general, the Distribution function express it like this:
So if you want to know, for example, the probability with which you will get a maximum of two hits, you have to add up the probabilities for 0 hit, 1 hit and 2 hits. “X”, in this case 2, stands for the highest probability. Based on the summation symbol, you insert 0, 1 and 2 for k and then add the probabilities for the result you are looking for.
Of course, the distribution function can also be plotted graphically. This graph shows the distributions for the case that 5 coins are tossed and the probability of success is 50%.
Binomial distribution example
A classic example of a binomially distributed random experiment is the drawing of balls from an urn, whereby, for example, drawing a red ball is rated as a success and drawing a black ball is rated as a failure. Instead of success or failure, one can also speak of hit and no hit.
Binomial distribution tasks
Below are more examples of tasks in the frame with binomial distributed random variables. For this tasks let n = 10 and given. In addition, the following applies: X is a binomially distributed random variable X. We are looking for the probability ...
1) ... for three successes / hits
2)... for a maximum of one hit
As under the paragraph Distribution function already explained, one has to add up the individual probabilities in the binomial distribution. For 2) we have added up the probability for P (X = 0) + P (X = 1). Alternatively, you can of course read the result from a distribution table, if available.
3)... at least one hit
Here we subtract 1 with the counter-probability. The big advantage that we can simply equivalent to that in exercise 2).
4)... more than one hit
Here, too, we work again, as in exercise 3), with a logical conversion into the counter-probability. We already know the probability of at most one hit from exercise 2).
5)... less than a hit
We already know the probability of a maximum of 0 hits from exercise 3.
Binomial distribution of descriptive stochastics
Below you will find an overview of the most important dimensions in connection with the Binomial distribution. This includes the Expected value, the Variance and the Standard deviation.
Binomial distribution expectation
The Expected value can be easily calculated using the following formula:
Multiply the number of draws by the probability of success and you get the expected value.
Binomial distribution variance
The formula, to calculate the Variance a binomially distributed random variable looks like this:
You can also calculate these simply by inserting the parameters n and p.
Standard deviation binomial distribution
The Standard deviation can be easily determined from the variance using the classic method.
The is therefore equal to the standard deviation.
The binomial coefficient n occurs over k in the binomial distribution. You don't know exactly how to calculate this coefficient anymore? No problem, in our video about the binomial coefficients we explain it again clearly! Check it out right now!
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