Solve this problem that involves a probability density function

In summary: Where does the 6 in the variance come from? Shouldn't it be a 2?Where does the 6 in the variance come from? Shouldn't it be a 2?True...let me amend that.##E(T^2)=-\int_0^{\frac{2}{3}} u^2⋅(3-\dfrac{9}{2}u )du####=-\int_0^{\frac{2}{3}} (3u^2-\dfrac{9u^3}{2} )du=-\left[u^3-\dfrac{9u^4}{8} \right]####=-\left
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
understanding of continous and discrete distribution
I am refreshing on this; ..after a long time...

Note that i do not have the solution to this problem.

1667393644541.png
I will start with part (a).

##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k##

it follows that,

##3k - \dfrac{3k}{2}=1##

##\dfrac{3k}{2}=1##

##k=\dfrac {2}{3}##

For part (b),

##E(T)=\int_0^{\frac{2}{3}} u⋅(3-\dfrac{9}{2}u )du=\left[\dfrac{3}{2}×\dfrac{4}{9}-\dfrac{3}{2}×\dfrac{8}{27}\right]=\dfrac{6-4}{9}= \dfrac{2}{9}##Ok let me know if that's correct...
 
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  • #2
Those both look OK to me. Do you know how to do the last two parts?

One minor  issue
chwala said:
##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k##
This is bad notation, since ##f## it's already defined to be the pdf, not the cdf.
 
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  • #3
Office_Shredder said:
Those both look OK to me. Do you know how to do the last two parts?

One minor  issue

This is bad notation, since ##f## it's already defined to be the pdf, not the cdf.
I will work on the last parts too...refreshing on this...should be fine ...Will post once done.
 
  • #4
Office_Shredder said:
Those both look OK to me. Do you know how to do the last two parts?

One minor  issue

This is bad notation, since ##f## it's already defined to be the pdf, not the cdf.
I thought pdf is generally defined to be ##f_{u}## being the derivative of the continuous distribution ##f(u)## ...different books have different notations/language that I pretty find to be time waster...my focus is on the concept...

Which is the right notation? Thanks @Office_Shredder
 
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  • #5
chwala said:
Which is the right notation? Thanks @Office_Shredder
In my stats class we used ##f_{X}(x)## for the PDF and ##F_{X}(x)## for the CDF for a given continuous random variable ##X##. I think that this is fairly standard, because most books and lecture notes I found online used it.
 
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  • #6
Office_Shredder said:
Those both look OK to me. Do you know how to do the last two parts?

One minor  issue

This is bad notation, since ##f## it's already defined to be the pdf, not the cdf.
##E(T^2)=\int_0^{\frac{2}{3}} u^2⋅(3-\dfrac{9}{2}u )du##

##=\int_0^{\frac{2}{3}} (3u^2-\dfrac{9u^3}{2} )du=\left[u^3-\dfrac{9u^4}{8} \right]##

##=\left[\dfrac{8}{27}-\dfrac{9}{8}×\dfrac{16}{81}\right]=\dfrac{8-6}{27}= \dfrac{2}{27}##

##⇒Var (T) = \dfrac{24-4}{81}=\dfrac{20}{81}##

...this part is easy and straightforward. Bingo!
 
  • #7
Where does the 24 in the variance come from? Shouldn't it be a 6?
 
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  • #8
Office_Shredder said:
Where does the 24 in the variance come from? Shouldn't it be a 6?
True...let me amend that.
 
  • #9
chwala said:
##E(T^2)=\int_0^{\frac{2}{3}} u^2⋅(3-\dfrac{9}{2}u )du##

##=\int_0^{\frac{2}{3}} (3u^2-\dfrac{9u^3}{2} )du=\left[u^3-\dfrac{9u^4}{8} \right]##

##=\left[\dfrac{8}{27}-\dfrac{9}{8}×\dfrac{16}{81}\right]=\dfrac{8-6}{27}= \dfrac{2}{27}##

##⇒Var (T) = \dfrac{6-4}{81}=\dfrac{2}{81}##

...this part is easy and straightforward. Bingo!
 

1. What is a probability density function (PDF)?

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

2. How do you solve a problem involving a probability density function?

To solve a problem involving a probability density function, you need to follow these steps:

  • Identify the random variable and its range of possible values.
  • Determine the probability density function for the random variable.
  • Use the properties of the probability density function to calculate the probability of a specific outcome or a range of outcomes.
  • Check your answer to ensure it is reasonable and makes sense in the context of the problem.

3. What is the difference between a probability density function and a probability mass function?

A probability density function is used for continuous random variables, while a probability mass function is used for discrete random variables. A probability density function gives the probability of a range of values, while a probability mass function gives the probability of a specific value.

4. Can a probability density function have a value greater than 1?

No, a probability density function cannot have a value greater than 1. This is because the total area under the curve of a probability density function must equal 1, representing the total probability of all possible outcomes.

5. How is a probability density function related to a cumulative distribution function?

A cumulative distribution function is the integral of a probability density function. It represents the probability that a random variable will take on a value less than or equal to a given value. In other words, the cumulative distribution function is the probability density function "cumulated" over a range of values.

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