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Likelihood
Common problems in probability theory refer to the probability of observations $x_1, x_2, ... , x_n$ given a certain model and given the parameters (let's call them $\theta$) involved. For instance the probabilities for specific situations in card games or dice games are often very straightforward.
However, in many practical situations we are ...
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Let's parse the notation and then answer your questions.
This "critical function" $\phi$ is a tool to make a decision. Given the value $X$ of a test statistic, independently observe a uniformly distributed variable $U$ (supported on $[0,1]$). Reject the null hypothesis when $U \le \phi(X);$ otherwise, do not reject it.
To be explicit, given an ...
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For starters, I want to make sure that I understand correctly that when you say "deriving," you mean "differentiating".
What you are asking is called "differentiating under the equal sign" and is covered in basic calculus classes. If you still have your calculus textbook, you might want to do a quick review before going on in machine learning. I don't mean ...
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Linear regression is about how $x$ influences/varies with $y$, the outcome or response variable. The model equation is
$$ Y_i =\beta_0 + \beta_1 x_{i1} + \dotsm + \beta_p x_{ip}+\epsilon_i
$$ say, and how $x$ is distributed doesn't by itself give information about the $\beta$'s. That's why your second form of likelihood is irrelevant, so is not used.
See ...
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Comments to the question indicate the interest focuses on random variables that are unbiased linear estimators. Despite this restriction, in order to draw conclusions about the full bivariate distribution of the two estimators (suitably standardized as in the CLT), you need some control over the coefficients in the linear combinations. In short, the answer ...
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If you only have the conditional probabilities $p(x|y)$ where $x$ is the predicted and $y$ the actual, you will also need the probability $p(y)$ of the actual. This way you can compute $p(x, y) = p(x|y) \cdot p(y)$ and from that you can get the MI.
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There seems to be a slight ambiguity in your problem proposition. It does not make sense to say 'how often do you have to try to make at least one basket'. Because as soon as there is one success the criterion is met and you would stop to sample new throw attempts, the question can be rephrased as 'how often do you have to try until the player makes the ...
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Maybe try out the negative binomial distribution:
https://en.wikipedia.org/wiki/Negative_binomial_distribution
It counts the number of successes until r failures are reached. Simply flip the probabilities of success and error in your case (and set r=1) and you have a match for your problem.
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This distribution is called the Gamma-Inverse Beta distribution in this paper. It is available in the R package brr.
nsims <- 1e6
alpha <- 3
beta <- 5
K <- 6
theta <- 4
sims <- rgamma(nsims, shape = K, rate = theta) * rbeta(nsims, alpha, beta)
plot(density(sims, to=3))
curve(brr::dGIB(x, K, beta, alpha, theta), # note that alpha and beta ...
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