Prove that you've got a probability density function

In summary, the conversation discussed the probability of a car starting up and not starting up, as well as the task of finding the Bernoulli probability function associated with testing cars until two functional ones are found. The attempt at a solution involved summing up probabilities and proving it to be a probability density function, but it was noted that a combinatorial factor was missing in the equation.
  • #1
pyro_peewee
2
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Homework Statement



Probability of a car starting up is 0.9
Probability of a car NOT starting up is 0.1

Cars are tested until 2 functional cars are found.

Find Bernoulli probability function associated and PROVE that it is a pdf (probability density function).

Homework Equations



?

The Attempt at a Solution



To prove that it is a pdf, I think that I need to sum up all of the probabilities associated with the x values and show that it equals 1.

[tex]\Sigma[/tex]x=2 to infinity p^2*(1-p)^(x-2) How do I show that this equals 1? Is it even correct?

Keep in mind that the minimum value for the # of cars tested is 2 because we're looking for 2 functional cars, and once we've got 2, we stop the trials.
 
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  • #2
It isn't a PDF. You are forgetting a combinatorial factor, aren't you? If you get a second success after x trials and quit, there is more than one way that the first success could have happened. Isn't there?
 
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What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the likelihood of a random variable taking on a specific value or falling within a certain range of values. It is often used in probability and statistics to model the distribution of data.

How do you prove that you have a probability density function?

To prove that a function is a probability density function, it must meet two criteria: it must be non-negative for all values of the variable, and the integral of the function over the entire space must equal one. This can be mathematically expressed as f(x) ≥ 0 for all x and ∫f(x)dx = 1.

What is the difference between a PDF and a cumulative distribution function (CDF)?

A PDF describes the probability of a random variable taking on a specific value, while a CDF describes the probability of a random variable being less than or equal to a given value. In other words, the PDF shows the likelihood of a single event occurring, while the CDF shows the likelihood of all events up to a certain value occurring.

Can a probability density function be used for discrete and continuous data?

Yes, a probability density function can be used for both discrete and continuous data. For discrete data, the PDF will take on a series of distinct values, while for continuous data, the PDF will be a smooth curve. In both cases, the integral of the PDF over the entire space will equal one.

Why is it important to have a probability density function?

A probability density function is important because it allows us to understand and analyze the distribution of data. It can help us make predictions and draw conclusions about the likelihood of certain events occurring. Additionally, it is used in many statistical and machine learning models to make accurate predictions and decisions based on data.

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