How to solve for probability given density function

In summary, the conversation discusses the probability density function of a random variable y and the question of finding the probability that 45y is less than or equal to 10. The solution involves computing the expected value and variance of y, using the cumulative distribution function, and solving for the probability by substituting a value for y.
  • #1
eric.mercer92
8
0

Homework Statement


The probability density function of a random variable y is:
f(y) = 100ye[itex]^{-10y}[/itex], if y>0
f(y) = 0 otherwise

What is the probability that 45y <= 10?



Homework Equations


E(y) = ∫yf(y)dy
Var(y) = ∫(y-E(y))f(y)dy



The Attempt at a Solution


I solved for the expected value of y and got E(y) = 0.2
I then got Var(y) = 0.02
I don't think this is a normal distribution because the PDF is not a normal
PDF. I don't know how to solve for the probability of 45y <= 10

Any help will be greatly appreciated!
 
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  • #2
Not quite sure, but maybe you do ∫f(y) dy from 0 to 10/45?
 
  • #3
I thought about that, but it just didn't seem right, after all, if one of the y's is over 10/45, the other y's can be less than that to make up for it.
 
  • #4
eric.mercer92 said:

Homework Statement


The probability density function of a random variable y is:
f(y) = 100ye[itex]^{-10y}[/itex], if y>0
f(y) = 0 otherwise

What is the probability that 45y <= 10?

Homework Equations


E(y) = ∫yf(y)dy
Var(y) = ∫(y-E(y))f(y)dy

The Attempt at a Solution


I solved for the expected value of y and got E(y) = 0.2
I then got Var(y) = 0.02
I don't think this is a normal distribution because the PDF is not a normal
PDF. I don't know how to solve for the probability of 45y <= 10

Any help will be greatly appreciated!

Homework Statement


Homework Equations


The Attempt at a Solution


You shouldn't be worrying about what sort of distribution this represents. You're already given a pdf. Compute the cdf (cumulative distribution function). If the pdf is [itex]f(y)[/itex], the cdf is [itex]\int_{-\infty}^y f(y)dy[/itex]. Remember that from -∞ to 0, f(y) = 0, so break it up into two integrals, one from -∞ to 0 (which vanishes), and the other from 0 to y.

Do the integration by parts and work out the definite integral in terms of y. Then it's as simple as substituting y = 10/45 into that.

The answer I get lies between 0.6 and 0.7, if you wish to check yours.
 
  • #5
Ahh yes, I see, thank you very much. I got the same answer.
Thanks again!
 
  • #6
eric.mercer92 said:
Ahh yes, I see, thank you very much. I got the same answer.
Thanks again!

No problemo. :smile:
 

FAQ: How to solve for probability given density function

What is a density function?

A density function is a mathematical function that describes the probability distribution of a random variable. It represents the relative likelihood of different outcomes occurring in a given range of values.

What is the relationship between probability and density function?

The probability of a random variable falling within a certain range of values can be calculated by integrating the density function over that range. In other words, the area under the density function curve represents the probability of the variable falling within that range.

How do I solve for probability given a density function?

To solve for probability, you can use the cumulative distribution function (CDF) which is the integral of the density function. The CDF gives the probability of the random variable being less than or equal to a specific value.

Can I use any density function to solve for probability?

No, the density function must meet certain criteria to be valid, such as being non-negative and integrating to 1 over its entire range. Examples of commonly used density functions include the normal distribution, binomial distribution, and exponential distribution.

Are there any limitations to using a density function to solve for probability?

Yes, density functions assume that the data follows a specific probability distribution. If the data does not follow this distribution, the results may not be accurate. Additionally, density functions may not be suitable for all types of data, such as discrete data or data with outliers.

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