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Results tagged with self-study
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user 24515
A routine exercise designed to test one's knowledge; often from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.
1
vote
Accepted
Independent in statistics and independent in linear algebra?
If two vectors are linearly independent, they are orthogonal, and their dot product is zero. The covariance is equivalent to (n times) the dot product, and the correlation is the normalized dot produc …
- 6,714
4
votes
Accepted
Confidence interval of the expected value
You could add the tag self-study to your question.
First estimate the expectation of the normal distribution by the sample mean, then the sample variance, and the square root of the sample variance. …
- 6,714
4
votes
2
answers
3k
views
Question about deriving posterior distribution from normal prior and likelihood
I am trying to understand how to derive the posterior distribution of a parameter $\mu$ given data vector $z$, $P(\mu|z)$, where
$$
\mu \sim N(0,A)
$$
and
$$
z|\mu \sim N(\mu,1).
$$
Obviously fro …
- 6,714
2
votes
Monte-Carlo simulation
A good starting point is https://en.wikipedia.org/wiki/Monte_Carlo_integration.
The goal is to find the volume (surface) under the curve $$f(x)=e^x-1$$ in the interval $\Omega=[0,1]$, which is equal …
- 6,714
9
votes
2
answers
12k
views
Covariance of two sample means
I am trying to derive the covariance of two sample means and get confused at one point. Given is a sample of size $n$ with paired dependent observations $x_i$ and $y_i$ as realizations of RVs $X$ and …
- 6,714
2
votes
0
answers
131
views
How to calculate integral of density under constraint?
Given a bivariate standard-normally distributed random variable $Y=[Y_1,Y_2]^T$ with density $\phi(Y_i,Y_2)$, the probability of $Y_1>0$ is simply $$P(Y_1 > 0)=1- \int_{-\infty}^{\infty} \int_{-\infty …
- 6,714
2
votes
Accepted
Examples where the evaluation of the posterior distribution $p(Z|X)$ of the latent variables...
Any method involving a latent variable can be represented in the terms you mentioned and the EM algorithm is often used as a part of the estimation or prediction procedure for $Z$, such as
Mixture m …
- 6,714
4
votes
How to derive the MLE of a Gaussian mixture distribution
I continued working on this exercise and came up with a solution. I'd be glad about comments.
Let $\theta=[\pi_0,\pi_1,\mu_0,\mu_1,\sigma_0^2,\sigma_1^2]$
The likelihood over N observations is giv …
- 6,714
7
votes
2
answers
8k
views
How to derive the MLE of a Gaussian mixture distribution
In my self-study, I consider a Gaussian mixture distribution:
$$p(x)= p(k=1) N(x|\mu_1,\sigma^2_1) + p(k=0) N(x|\mu_0,\sigma^2_0)$$
where $p(k=1)+p(k=0)=\pi_1+\pi_0=1$. I am now asked to do three th …
- 6,714