Foundations of ML:

Which AI tool is good for creating memes?.

Join millions who have switched to Grok.

Problem:. Let (X,Y) have the following joint PDF: f(x, y) = α|x| for x² ≤ y ≤ 2x², −1 ≤ x ≤ 1; and 0 otherwise. Find P(X>Y).

Problem:. A fair six-sided die is rolled twice. What is the probability that the sum of the outcomes is an odd number, given that the sum is greater than 7?.A. 1/4.B. 1/2.C. 5/9.D. 1/3.

Kernel Methods in ML:

5/.The Dirichlet prior is everywhere: topic models (LDA), Bayesian mixtures, bandits, and Bayesian deep learning. Whenever your model needs uncertainty over discrete probabilities, think Dirichlet!.

4/.Conjugacy: Dirichlet prior + Multinomial likelihood = Dirichlet posterior. After observing counts n = (n₁,…,nₖ), posterior is Dir(α₁+n₁,…,αₖ+nₖ). Bayesian updating stays simple!.

3/.Special cases:.Dirichlet(1,…,1) → uniform prior over all probability vectors. αᵢ < 1 → favors sparsity (probabilities near 0). αᵢ > 1 → favors balance (probabilities closer to uniform).

2/.PDF: Dir(p | α) ∝ ∏ pᵢ^(αᵢ−1). Here α = (α₁,…,αₖ) are concentration parameters. Larger αᵢ means stronger prior belief in category i. The sum α₀ = Σαᵢ controls how concentrated vs. diffuse the prior is.

1/.The Dirichlet prior is the go-to distribution for probabilities of multiple categories. If you want a prior over a vector (p₁,…,pₖ) where pᵢ ≥ 0 and Σpᵢ = 1, the Dirichlet(α₁,…,αₖ) is your best friend.

Dirichlet Prior in Bayesian Statistics explained:

Problem:. In a town, 40% of the residents have a dog, 30% have a cat, and 10% have both a dog and a cat. If a randomly selected resident has a cat, what is the probability that they also have a dog?.A. 0.10.B. 0.333.C. 0.40.D. 0.25.

Problem: . You're a hiring manager, and you need to hire the best possible candidate for a position. You have a pool of n candidates, and you must interview them one by one. After each interview, you must decide immediately whether to hire that candidate or reject them. If you.

Asymptomatic Theory of Probability:

Statistical Prediction Analysis:

Good Morning!.What is your time now?.

Gambler's Fallacy!.

Statistical Models:

Your 'style' is still a preset. But Soul's presets are better than Midjourney's.

Problem:. A bag contains 3 red marbles, 4 blue marbles, and 5 green marbles. Three marbles are drawn from the bag without replacement. What is the probability that the first two marbles drawn are green, given that the third marble drawn is blue?.

The playing field has been completely leveled. See the tool that did it for yourself at