Solve the problem involving probability density function

In summary: F(4)\geq 0## and ##F(x)## is not continuous at ##x=1##.In summary, the answer is no, substituting the upper limit will not give you the correct PDF.
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
Statistics
This is the question:

1663419354750.png
This is the ms solution- from Further Maths paper.

1663419458348.png
My question is referenced to the highlighted part. I can see they substituted for the lower limit i.e ##x=1## to get: ##F(x)=\dfrac{x^3-1}{63}##
supposing our limits were; ##2≤x≤4## would the same approach apply? Anything wrong if we substitute the upper limit i.e ##x=4##? and have our:
##F(x)=\dfrac{64-x^3}{63}?##
 
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  • #2
I don't understand your question.

If [itex]f[/itex] is to be a PDF for a random variable [itex]X[/itex] taking values in [itex][a,b][/itex] we must have [tex]\int_a^b f(x)\,dx = 1.[/tex] If that is the case, then for [itex]a \leq c < d \leq b[/itex] we have [tex]
P(c \leq X \leq d) = \int_c^d f(x)\,dx.[/tex]
 
  • #3
pasmith said:
I don't understand your question.

If [itex]f[/itex] is to be a PDF for a random variable [itex]X[/itex] taking values in [itex][a,b][/itex] we must have [tex]\int_a^b f(x)\,dx = 1.[/tex] If that is the case, then for [itex]a \leq c < d \leq b[/itex] we have [tex]
P(c \leq X \leq d) = \int_c^d f(x)\,dx.[/tex]
That's clear...my question is suppose we substitute the upper limit ; ##x=4## instead of lower limit; ##x=1## to get ##F(x)##, would that be correct? Note that when we integrate ##f(x)##, we shall get the indefinite integral,##F(x)##= ##\dfrac{x^3}{63}##.
 
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  • #4
By definition, [tex]
F(x) = P(X \leq x) = \int_1^x f(x)\,dx.[/tex] Not integrating from 1 to [itex]x[/itex] will not give you [itex]F(x)[/itex].
 
  • #5
pasmith said:
By definition, [tex]
F(x) = P(X \leq x) = \int_1^x f(x)\,dx.[/tex] Not integrating from 1 to [itex]x[/itex] will not give you [itex]F(x)[/itex].
Noted thanks @pasmith. Let me look at this again. The concept is clear, was just wondering if we could reverse the order but that does not seem to work going with the general definition of probability density function.
 
  • #6
What of: $$-\int_x^1 f(x)dx=\int_1^x f(x)dx?$$ in our case
$$-\int_ x^1 \left[\dfrac{x^2}{21}\right] dx=-\int_ 4^x \left[\dfrac{x^2}{21}\right] dx=-\left[\dfrac{x^3-64}{63}\right]?$$
 
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  • #7
You're missing a 1- when you switch the bound from 1 to 4 since you're computing the complement probability.
 
  • #8
Office_Shredder said:
You're missing a 1- when you switch the bound from 1 to 4 since you're computing the complement probability.
I am not getting you...what I wanted to know is, "will my ##F(x)## in post ##6## be correct?' Having changed the bounds...
 
  • #9
No. The last line starts by saying ##P(X < x) = P(X> x)##. It's not though, you should use ##P(X < x) =1- P(X> x)##

You can verify you got the wrong answer by seeing that for you, ##F(1)\neq 0##
 
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1. What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a specific value. It is used to represent the probability distribution of a continuous random variable.

2. How is a probability density function (PDF) different from a probability mass function (PMF)?

A probability density function (PDF) is used to represent the probability distribution of a continuous random variable, while a probability mass function (PMF) is used for discrete random variables. A PDF assigns probabilities to intervals of values, while a PMF assigns probabilities to specific values.

3. How do you calculate the probability from a probability density function (PDF)?

To calculate the probability from a probability density function (PDF), you need to integrate the function over a specific interval of values. The resulting value represents the probability of the random variable falling within that interval.

4. What is the relationship between a probability density function (PDF) and a cumulative distribution function (CDF)?

The cumulative distribution function (CDF) is the integral of the probability density function (PDF). It represents the probability that a random variable takes on a value less than or equal to a specific value. The CDF can be used to calculate probabilities for a continuous random variable, while the PDF is used to represent the probability distribution.

5. How is a probability density function (PDF) used in real-world applications?

A probability density function (PDF) is used in a variety of fields, including statistics, physics, and engineering. It is used to model and analyze data, make predictions, and make decisions based on probabilities. For example, in finance, PDFs are used to model stock prices and predict future market trends.

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