Density/Distribution Functions

[SGS]

In my statistics class I missed the lecture on this stuff. Our book gives hardly any examples and there is a mid-term tomorrow. I am meeting w/ my professor, but not til after the mid-term. Any of you have any websites that give info or you personally have info on how to do these problems? For instance a problem could be:

Suppose X has a density function f(x)=4x^3 for 0<x<1, 0 otherwise. Find the distribution function and P(X<1).

Basically I am just asking how you start these problems and the procedure you go through to complete them. Any help is appreciated. Thanks in advance.

SGS

[BruceZ]

[ QUOTE ]
In my statistics class I missed the lecture on this stuff. Our book gives hardly any examples and there is a mid-term tomorrow. I am meeting w/ my professor, but not til after the mid-term. Any of you have any websites that give info or you personally have info on how to do these problems? For instance a problem could be:

Suppose X has a density function f(x)=4x^3 for 0<x<1, 0 otherwise. Find the distribution function and P(X<1).

Basically I am just asking how you start these problems and the procedure you go through to complete them. Any help is appreciated. Thanks in advance.

SGS

[/ QUOTE ]

The distribution function is just the integral of the density function from minus infinity to t, and it tells you the probability that X lies between minus infinity and t, or P(X<=t). Replace the integral with a sum for a discreet density function.

[LetYouDown]

I'm curious how this function is practically used. From your description, I'm assuming the distribution function for the example density function is x^4 from -infinity to t. Are there any simple, "real world" examples of these functions in use?

[BruceZ]

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I'm curious how this function is practically used. From your description, I'm assuming the distribution function for the example density function is x^4 from -infinity to t.

[/ QUOTE ]

F(t) = t^4 for 0 <= t <= 1
F(t) = 0 for t <= 0
F(t) = 1 for t >= 1



[ QUOTE ]
Are there any simple, "real world" examples of these functions in use?

[/ QUOTE ]

Sure, lots of them. For example, your long term poker winnings are described by a normal or Gaussian density function (bell curve), and the normal distribution function tells you the probability that your winnings will be less than some amount. In that case the density function is 1/[sqrt(2*pi)*sigma]*exp[-(x-u)^2/2sigma^2] where u is your average win and sigma is the standard deviation of the win. The distribution function obtained by integrating this numerically is available in tables.

[carsten]

The distribution function D for a random variable X can be used to determine the probability that X lies between any two given bounds Xmin and Xmax by taking the difference D(Xmax)-D(Xmin).

Note that this is just a shortcut for computing the definite integral of the density function between the bounds Xmin and Xmax, since the distribution function is simply an antiderivative of the density function.

-Carsten