Top Mathematics discussions
NishMath
|
Coffeeman@Recent Questions - Mathematics Stack Exchange - 36d
Recent mathematical research has focused on the fascinating properties of topological spaces, particularly examining how curves behave when lifted from a torus to the Euclidean plane. A key finding confirms that if a closed curve on a torus is simple (meaning it does not intersect itself), its straight-line representative in the plane is also simple. This is particularly relevant in mapping class groups, where understanding the geometry of curves in this way is important for further analysis.
Furthermore, investigations have explored the conditions under which a Tychonoff space remains sequentially closed within its Stone-Čech compactification. It was determined that if every closed, countable, discrete subset of the space is C*-embedded, then the space is sequentially closed in its Stone-Čech compactification. This result provides tools for characterizing spaces which have this property. Researchers have also studied the nature of almost discrete spaces, seeking examples and characterizations within topological theory, and relating to properties like C-embeddedness and separation of sets.
Share:
References :
Classification:
Will@Recent Questions - Mathematics Stack Exchange - 25d
A recent analysis has delved into the probabilistic interpretation of linear regression coefficients, highlighting the differences in reasoning when using expected values versus covariances. It has been shown that when calculating regression coefficients, employing expected values leads to correct formulations that correspond to the ordinary least squares (OLS) method. Specifically, the formula a=E[XY]/E[X^2] is derived using the expected value of the product of the independent and dependent variables. This approach aligns with the traditional understanding of linear regression where a model is expressed as Y=aX+ε, with ε being a centered error term independent of X.
However, using covariances for the probabilistic interpretation fails, especially in models without an intercept term. While covariance is often used to calculate the correlation between variables, the derived formula a=cov(X,Y)/var(X) does not align with the correct regression coefficient when there isn't an intercept. This divergence arises because the assumption of an intercept is implicit when using covariance, and its absence invalidates the formula using covariance. The study clarifies how formulas are derived in both scenarios and why the probabilistic reasoning fails when using covariances in situations where there is no intercept included in the model. The use of empirical means versus population means was also discussed to explore the nuances further.
Share:
References :
Classification:
@medium.com - 18d
Statistical distributions and their applications are crucial in understanding data and making informed decisions. One common application is the Chi-squared test used to evaluate if a Linear Congruential Generator (LCG) produces random numbers that follow a uniform distribution. A key point of discussion revolves around the interpretation of the p-value in this test; with a small p-value, typically less than 0.05 indicating a low probability of the data conforming to the expected distribution, leading to the rejection of the hypothesis. This contrasts with an earlier misunderstanding where some had thought a small p value means the data follows the desired distribution more closely.
Another area is binomial distribution, which is used when dealing with experiments that have two possible outcomes. This distribution can be applied to scenarios like predicting sales success based on the probability of closing a deal with each sales call. In these cases, tools like Microsoft Excel can be used to calculate the likelihood of achieving different numbers of successful sales within a fixed number of calls. The binomial and Poisson distributions are also very important in probability and statistics, with the binomial distribution counting the number of successes in a fixed number of independent trials, while the Poisson distribution models the probability of a number of events occurring within a fixed time or space. These distributions are fundamental to probability theory and are frequently used in various practical situations and are also easy to model using Python for ease of understanding.
Share:
References :
Classification:
|
|