How to Calculate Probability Density Functions for Exact Numbers

In summary, the conversation discusses how to calculate the probability of a specific value for a continuous random variable with a continuous probability density function. In question (ii), it is mentioned that it is a trick question and that the probability will be the same for all continuous random variables. In question (iii), the definition of expected value is asked and it is also stated that integration by parts is used to solve the equation. The conversation ends with a solution being suggested using a simple u-substitution.
  • #1
t_n_p
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Homework Statement


http://img204.imageshack.us/img204/2097/34629164kd2.jpg [Broken]

The Attempt at a Solution


I know how to compute something like Pr(x<0.25) for example, but I'm unsure how to do it for an exact number like in question (ii). I attempted to integrate and then sub x=1/4 where neccisary, but to no avail!

Part (iii) I really have no idea!

Would be grateful is someone could explain these to me. Thanks! :cool:
 
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  • #2
ii) Is a trick question in a sense. Hint: the P(X=1/4) is the same for all continuous random variables with a continuous pdf.

iii) What is the definition of expected value?
 
  • #3
Ok, so P(x=1/4)=0 for all condinuous random variables with a continuous pdf, how come?

Re part (iii) straight swap (cos(pi*x)) for X into this equation?
http://img233.imageshack.us/img233/5191/92698641bo2.jpg [Broken]
If I do that, i get a nasty integral of cos(pi*x)*sin(pi*x)dx

Pardon my silly questions, I'm rusty as hell..
 
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  • #4
For any continuous function f, what is

[tex]\lim_{\epsilon\to 0}\int_{a-\epsilon}^{a+\epsilon}f(x)dx[/tex]Re part (iii). This is a simple integral. If the factor of pi is throwing you off, think of this as

[tex]\int \cos(ax)\sin(ax)dx[/tex]

and then set [itex]a=\pi[/itex] after integrating.
 
  • #5
Ah ha, got you for part (ii), it's not just 1/4 then, it's any number correct!?

re (iii), I'm still bloody lost. tried integration by parts, but the integration part just returns the same integral so it's like going around in a circle (if you know what i mean). I can't think of any other way to solve it :confused:
 
  • #6
Try a simple u-substitution.
 
  • #7
Let u= sin([itex]\pi[/itex]x).
 
  • #8
Ah gotcha!
 

What is a probability density function?

A probability density function (PDF) is a mathematical function that describes the distribution of a continuous random variable. It shows the relative likelihood of a random variable taking on a particular value within a given range.

How is a probability density function different from a probability distribution function?

A probability distribution function (PDF) is a function that assigns probabilities to discrete outcomes of a random variable. A probability density function (PDF), on the other hand, is used for continuous random variables and represents the density of the probability distribution at a particular point.

What is the difference between a probability density function and a cumulative distribution function?

A probability density function (PDF) shows the relative likelihood of a random variable taking on a specific value, while a cumulative distribution function (CDF) shows the probability of a random variable being less than or equal to a certain value. In other words, the PDF gives the probability density at a particular point, while the CDF gives the cumulative probability up to that point.

How do you interpret the area under a probability density function curve?

The area under a probability density function (PDF) curve gives the probability of a random variable falling within a specific range of values. This area represents the total probability of all possible outcomes within that range.

Can a probability density function have negative values?

No, a probability density function (PDF) cannot have negative values. The values of a PDF must be non-negative, as they represent the relative likelihood of a random variable taking on a particular value. However, the area under the PDF curve can be negative if the curve falls below the x-axis in certain regions.

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