Problem involving Probability density function

In summary, the conversation discusses the equivalence of the notations < 0.5 and ≤ 0.5 in continuous probability distributions. The point 0.5 is said to have zero width or measure, making the two notations equivalent. The conversation also mentions finding the cdf at x=0.5 and correcting a typo in the notation for the integral.
  • #1
chwala
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Homework Statement
see attached
Relevant Equations
stats
1648817667989.png


I just want to be certain, i think the inequality indicated is not correct...ought to be less than. Kindly confirm...This is a textbook literature.
 
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  • #2
In terms of continuous probability distributions ##< 0.5## and ##\le 0.5## are equivalent, because the point ##0.5## itself has zero width (or zero measure if you prefer).
 
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  • #3
PeroK said:
In terms of continuous probability distributions ##< 0.5## and ##\le 0.5## are equivalent, because the point ##0.5## itself has zero width (or zero measure if you prefer).
Thanks Perok, so the pdf indicated above is just the same as finding the cdf at ##x=0.5## right? giving us ##F(0.5)=0.125##.
 
  • #4
chwala said:
Thanks Perok, so the pdf indicated above is just the same as finding the cdf at ##x=0.5## right? giving us ##F(0.5)=0.125##.
I think so. I haven't looked very carefully at the material you posted.
 
  • #5
1648818911263.png


ought to be integral of ##f(x)## and not ##x##... or is it fine the way it is?
 
  • #6
chwala said:
View attachment 299242

ought to be integral of ##f(x)## and not ##x##... or is it fine the way it is?
Isn't it obvious that's a typo?
 
  • #7
ok cheers Perok.
 

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

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

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

A PDF is used for continuous random variables, while a PMF is used for discrete random variables. This means that a PDF represents the probabilities of an infinite number of possible outcomes, while a PMF only represents the probabilities of a finite number of possible outcomes.

3. How do you calculate the area under a PDF curve?

The area under a PDF curve represents the probability of a random variable falling within a certain range. To calculate it, you would need to integrate the PDF function over that range. This can be done using calculus.

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

A CDF is the integral of a PDF and represents the probability of a random variable being less than or equal to a given value. In other words, the CDF is the cumulative sum of all the probabilities in the PDF up to a certain point.

5. How can a PDF be used in real-world applications?

A PDF can be used to model and analyze various real-world phenomena, such as stock prices, weather patterns, and population growth. It is also used in fields such as statistics, economics, and engineering to make predictions and inform decision-making processes.

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