Three Questions on Interpreting the Outcome of the Probability Density Function I have three questi

Armeninilu 2022-06-11 Answered
Three Questions on Interpreting the Outcome of the Probability Density Function
I have three questions: What is the interpretation of the outcome of the Probability Density Function (PDF) at a particular point? How is this result related to probability? What we exactly do when we maximize the likelihood?
To better explain my questions:
(i) Consider a continuous random variable X with a normal distribution such that μ = 1.5 and σ 2 = 2. If we evaluate the PDF at a particular point, say 3.4, using the formula:
f ( X ) = 1 2 π σ 2 e ( X μ ) 2 2 σ 2   ,
we get f ( 3.4 ) = 0.1144. How we interpret this value?
(ii) I previously read that the result of 0.1144 is not necessarily the probability that X takes the value of 3.4. But how the result is related to probability concept?
(iii) Consider a sample of the continuous random variable X of size N=2.5, such that X 1 = 2 and X 2 = 3.5. We can use this sample to maximize the log-likelihood:
max ln L ( μ , σ | X 1 , X 2 ) = ln f ( X 1 ) + ln f ( X 2 )
If f(X) is not exactly a probability, what are we maximizing? Some texts detail that "we are maximizing the probability that a model (set of parameters) reproduces the original data". Is this phrase incorrect?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Leland Ochoa
Answered 2022-06-12 Author has 25 answers
I'm not an expert, but this is the way I understand it. Denote the cumulative distribution function (CDF) by
F X ( x ) = x f X ( t )   d t .
(i) For small ε,
f X ( x ) F X ( x + ε ) F X ( x ) ε ,
so
F X ( x + ε ) F X ( x ) + ε f X ( x ) .   ( )
That is, the PDF f X ( x ) gives the "rate of change" of the CDF F X ( x ).
To illustrate using your example, we can approximate F X ( 3.401 ) using (∗) above. We get
F X ( 3.401 ) F X ( 3.4 ) + 0.001 f X ( 3.4 ) = 0.910445404 + 0.001 0.114404814 = 0.910559808.
Using Excel, we see that F X ( 3.401 ) = 0.910559754, which is very close to our approximation. (I also used Excel to compute F X ( 3.4 ).)
(ii) As you said, f X ( 3.4 ) is not the probability that X=3.4. In fact, the probability that X=3.4 if X is a continuous random variable is 0 since
3.4 3.4 f X ( t )   d t = 0.
In general, the probability that X=x, where x is a real number, is 0.
Not exactly what you’re looking for?
Ask My Question
Tristian Velazquez
Answered 2022-06-13 Author has 7 answers
The probability density function is the unsigned derivative of the cumulative probability function.
f X ( x ) = | d     d x P ( X x ) |
It may be considered the "gradient of the tangent" of the curve; that is, the "rate of change" of accumulation of probability, as value for the continuous random variable increases.
So you are maximising the amount the parameters contribute to immediate accumulation of probability around the data points.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-02-23
Interpreting z-scores: Complete the following statements using your knowledge about z-scores.
a. If the data is weight, the z-score for someone who is overweight would be
-positive
-negative
-zero
b. If the data is IQ test scores, an individual with a negative z-score would have a
-high IQ
-low IQ
-average IQ
c. If the data is time spent watching TV, an individual with a z-score of zero would
-watch very little TV
-watch a lot of TV
-watch the average amount of TV
d. If the data is annual salary in the U.S and the population is all legally employed people in the U.S., the z-scores of people who make minimum wage would be
-positive
-negative
-zero
asked 2022-06-27
How do you solve x - 2 x + 3 < x + 1 x ?
asked 2022-06-20
Problem involved comparing two variance
The body mass index (𝐵𝑀𝐼) is a factor in assessing the health of a person. The data given are the 𝐵𝑀𝐼 for random samples of 18 women and 20 men selected in a town. Assume that the 𝐵𝑀𝐼 for such women are normally distributed with variance 25, while the 𝐵𝑀𝐼 for such men are normally distributed with the variance of 16. Construct a 96% confidence interval for the difference between the mean 𝐵𝑀𝐼 of the women and men in the town. Interpret your results.
The above is the Question given, from my understanding the variance given is the sample variance but which test should I use because I don't know whether the population variance is equal or unequal.
The question has provided a set of data but didn't mention it is population data or sample data and request to pick the data randomly from the dataset.
asked 2022-06-27
Setting We work on a filtered probability space ( Ω , F , ( F ) t [ 0 , T ] , P ). Let ( X n ) n and X be finite variation processes such that X n converges pointwise to X, i.e.
lim n X t n ( ω ) = X t ( ω )
for all ( ω , t ) Ω × [ 0 , T ], and such that ( X n ) n converges pointwise to X (where t denotes the total variation on [0,t]).
Question Does this imply that sup n X n < pointwise?
asked 2021-11-16

Use a direct proof to show that the sum of two even integers is even.

asked 2022-07-24
Find the standard form of the equation with endpoints of a diameter at points (1,4) and (-5,8)
asked 2021-05-19
For digits before decimals point, multiply each digit with the positive powers of ten where power is equal to the position of digit counted from left to right starting from 0.
For digits after decimals point, multiply each digit with the negative powers of ten where power is equal to the position of digit counted from right to left starting from 1.
1) 100=1
2) 101=10
3) 102=100
4) 103=1000
5) 104=10000
And so on...
6) 101=0.1
7) 102=0.01
8) 103=0.001
9) 104=0.0001

New questions