Understanding the concept of Probability distribution

In summary, the problem appears to be that when solving for ##F_X(u)## using the given probability density function, the value of ##F_X(u)## is equal to ##1## when ##u=b##. This is a mistake because the correct value of ##F_X(u)## is ##\dfrac{1}{b-a}##.
  • #1
chwala
Gold Member
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Homework Statement
see attached
Relevant Equations
statistics
Consider the attachment below;

1667555414361.png


How did they arrive at

##F_X (u) = \dfrac{u-a}{b-a}## ?

I think there is a mistake on the inequality, probably its supposed to be ##a≤u<b## and that will mean;

$$F_X (u) =\dfrac{1}{b-a} \int_a^u du= \dfrac{1}{b-a} ⋅(u-a)$$ as required. Your thoughts...then i have the second part of the question that i will post here after the analysis.
 
Last edited:
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  • #2
What do you think goes wrong if you include b? What is ##F(b)## supposed to be, and what does it become?
 
  • #3
Office_Shredder said:
What do you think goes wrong if you include b?
It will be equal to ##1##.
 
  • #4
chwala said:
It will be equal to ##1##.
And why would that be a problem?

Do note that the definition of ##F_X## would result in exactly the same function if you instead defined the middle case on the interval ##[a,b)## and the third on ##[b, \infty)## because both expressions are equal to one at ##u = b##.
 
  • #5
Orodruin said:
And why would that be a problem?

Do note that the definition of ##F_X## would result in exactly the same function if you instead defined the middle case on the interval ##[a,b)## and the third on ##[b, \infty)## because both expressions are equal to one at ##u = b##.
Ok, i hear you...then are my steps in post ##1## correct with the given limits? if not then how did they arrive at the solution.
 
  • #6
Orodruin said:
And why would that be a problem?

Do note that the definition of ##F_X## would result in exactly the same function if you instead defined the middle case on the interval ##[a,b)## and the third on ##[b, \infty)## because both expressions are equal to one at ##u = b##.
My intention is to get the shown ##F_X(u)## from the given probability density function ##f_x(u)=(b-a)^{-1}##. This is where my problem is.
 
  • #7
1667568416160.png
This is the continuation of the same problem...clearly textbook mistake on ##E(x)## value i.e the highlighted in red...Anyway, my approach on this problem;

$$E(x)=\dfrac{1}{b-a} \int_a^b u du= \left[\dfrac{b^2-a^2}{2} ⋅\dfrac{1}{b-a}\right]=\left[\dfrac{(b+a)(b-a)}{2(b-a)}\right]=\dfrac{b+a}{2}$$$$Var(x)=\dfrac{1}{b-a} \int_a^b u^2 du=\left[\dfrac{b^3-a^3}{3} ⋅\dfrac{1}{b-a}\right]=\left[\dfrac{b^2+a^2+ab}{3}-\dfrac{(b+a)^2}{4}\right]$$

$$=\left[\dfrac{(4b^2+4a^2+4ab)-(3b^2+6ab+3a^2)}{12}\right]$$

$$=\left[\dfrac{b^2+a^2-2ab}{12}\right]= \dfrac{(b-a)^2}{12}$$

if there is a better approach will appreciate. Cheers guys!
 
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1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of different outcomes occurring in a random experiment or event. It assigns a probability to each possible outcome, and the sum of all probabilities is equal to 1.

2. What is the difference between discrete and continuous probability distributions?

A discrete probability distribution is one where the possible outcomes are countable and have a finite or countably infinite number of values. Examples include flipping a coin or rolling a dice. A continuous probability distribution is one where the possible outcomes are uncountable and have a range of values. Examples include measuring the height or weight of individuals.

3. How is a probability distribution represented?

A probability distribution can be represented in various ways, such as a table, graph, or mathematical equation. The most common graphical representation is a histogram, where the bars represent the probability of each outcome. The mathematical equation used to represent a probability distribution depends on the type of distribution, such as the binomial, normal, or Poisson distribution.

4. What is the importance of understanding probability distributions in science?

Probability distributions are essential in science because they allow us to make predictions and draw conclusions about the likelihood of certain outcomes. They are used in various fields, such as statistics, physics, biology, and finance, to analyze data and make informed decisions. Understanding probability distributions also helps in designing experiments and interpreting results accurately.

5. How can we use probability distributions in real-life situations?

Probability distributions are used in real-life situations to make decisions and assess risk. For example, insurance companies use probability distributions to calculate premiums, and businesses use them to make financial projections. In everyday life, we use probability distributions to make decisions, such as whether to carry an umbrella based on the probability of rain or to invest in a stock based on its historical performance.

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