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Update probability_theory.Rmd #25

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Update probability_theory.Rmd
erocarrera Dec 29, 2019
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Update probability_theory.Rmd
erocarrera Jan 20, 2020
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10 changes: 5 additions & 5 deletions probability_theory/probability_theory.Rmd
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Expand Up @@ -719,7 +719,7 @@ reparameterizations then translate from one equivalent manifestation to another.
For discrete spaces all reparameterizations can be reduced to permutations that
simply rearrange the individual elements of the space. This simple intuition,
however, doesn't carry over to continuous spaces for which reparameterizations
can be signficantly more subtle.
can be significantly more subtle.

# Great Expectations {#sec:expectation_values}

Expand Down Expand Up @@ -930,7 +930,7 @@ $$
\mathbb{m}_{\pi} = \mathbb{m}_{\pi, 1} = \mathbb{E}_{\pi} [ \iota ],
$$
which quantifies a sense of _where_ in the target space the probability
distribution is concentration its allocation.
distribution is concentrating its allocation.

At the same time the _variance_ of a probability distribution is the second
central moment,
Expand Down Expand Up @@ -1250,7 +1250,7 @@ poisson_prob(A1, l)
which delightedly yield the same answer.

Given this rousing success we can then ask what is the probability allocated to
the compliment of $A_{1}$?
the complement of $A_{1}$?
```{r}
plot_poisson(l)
plot_poisson_probs(0:(A1[1] - 1), l)
Expand Down Expand Up @@ -1583,7 +1583,7 @@ norm_prob <- function(B_min, B_max, mu, sigma) {
norm_prob(B1_min, B1_max, mu, sigma)
```

And the probability allocated to the compliment of $B_{1}$?
And the probability allocated to the complement of $B_{1}$?
```{r}
plot_norm(mu, sigma)
plot_norm_probs(mu, sigma, -8, B1_min)
Expand Down Expand Up @@ -2095,7 +2095,7 @@ indicator <- function(x, A) {
```

For example, to compute the probability of the set $A_{1}$ we evaluate the
corresponding indictor function at each of our samples and then compute the
corresponding indicator function at each of our samples and then compute the
Monte Carlo estimator and standard error,
```{r}
pushforward_samples = sapply(stan_samples, function(x) indicator(x, A1))
Expand Down