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Archive for the ‘Statistics’ Category

Below is the multinomial formula used in multinomial distribution in statistics (probabilidad multinomial). To explain what a multinomial distribution or multinomial probability distribution is, let’s suppose there are n objects. We put the n objects into boxes with each box having k objects. Simple math shows that we need n/k boxes.

Multinomial formula used in multinomial distribution problems

What is the multinomial formula?

The multinomial formula shown above is the number of ways to split the objects (n objects) up into boxes. The multinomial formula assumes that it makes a difference what box each group is put into.

The general form of Multinomial formula

In general the multinomial formula says that if you are putting n objects in m groups with k₁ objects in the first group, k₂ objects in the second, etc… and k_m objects in group m, then there are:

ways of doing this. Bear in mind that:

Relationship between multinomial formula and combinations formula

When m = 2, the multinomial formula becomes the same as the combinations formula.

In this section of Math Help online, we discuss L’hopital’s Rule which nicely follows Rolle’s Theorem and Mean Value Theorem. 

What is L’Hospital’s Rule?

L’Hopital’s Rule or L’Hospital’s Rule (sometimes just L’Hopital) is a rule permitting the evaluation of the limit of an indeterminate quotient of functions as the quotient of the limits of their derivatives. For example, for sin x, 

 

is an indeterminate of the form 0/0 but it can be evaluated as 

 

Definition of L’Hopital’s Rule

L’Hospital’s Rule is an application of the Mean Value Theorem and lies in the evaluation of 

 

where f(a) = 0 and g(a) = 0. 

The L’Hopital Rule states that if the 

 

is an indeterminate form (such as 0/0), then we can differentiate the numerator and the denominator separately and arrive at an expression that has the same limit as the original problem. Thus, 

 

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Who invented L’Hopital’s Rule?

L’Hopital’s Rule was named after the French Math analyst and geometer, Guillaume Francois Antoine de l’Hopital, Marquis de St Mesme (1661 – 1704). Guillaume Francois Antoine de l’Hopital was the author of the first textbook on differential calculus, but Guillaume Francois Antoine de l’Hopital is believed to have bought the rights to this rule from its discoverer.

In the last section of Math Help online, we covered Rolles’ theorem. Rolle’s Theorem has many uses in Mathematical analysis. In this section of Math Help, we discuss Mean Value Theorem which follows on from Rolles’ theorem. 

Definition of Mean Value Theorem

Mean Value Theorem is an elementary math result in mathematical analysis due to Lagrange that states that: 

If a real function is continuous at every point on a closed interval [a,b] and differentiable (or has a derivative at every point) on the open interval (a,b), then there is at least a point in the open interval c at which the first derivative of the function f ‘ (c) equals 

f(b) – f(a) 

b – a 

Therefore, there is a point on any arc of the graph of the function at which the tangent is parallel to the chord joining the end points of the arc. 

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The Generalized Mean Value Theorem or Cauchy’s Mean Value Theorem

The generalized mean value theorem known as Cauchy’s mean value theorem extends this to show that given two such functions, f and g, one can solve 

f ‘(c) [ g(b) - g(a) ] = g ‘(c) [ f(b) - f(a) ] 

for some c in [a,b].