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Edit typos in prob_meaning.md #211

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Edit typos in prob_meaning.md
natashawatkins Mar 9, 2022
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Update prob_meaning.md
natashawatkins Mar 9, 2022
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10 changes: 5 additions & 5 deletions lectures/prob_meaning.md
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Expand Up @@ -78,7 +78,7 @@ The random variable $X $ takes on possible values $k = 0, 1, 2, \ldots, n$ wit

$$
\textrm{Prob}(X = k | \theta) =
\left(\frac{n!}{k! (n-k)!} \right) \theta^k (1-\theta)^{n-k} =
\left(\frac{n!}{k! (n-k)!} \right) \theta^k (1-\theta)^{n-k}
$$

where the fixed parameter $\theta \in (0,1)$.
Expand All @@ -95,7 +95,7 @@ Here

Consider the following experiment:

Take $I$ **independent** sequences of $n$ **independent** flips of the coin**
Take $I$ **independent** sequences of $n$ **independent** flips of the coin

Notice the repeated use of the adjective **independent**:

Expand Down Expand Up @@ -333,7 +333,7 @@ as $I$ goes to infinity.

## Bayesian Interpretation

We again a binomial distribution.
We again use a binomial distribution.

But now we don't regard $\theta$ as being a fixed number.

Expand Down Expand Up @@ -638,7 +638,7 @@ $$
={Beta}(\alpha + k, \beta+N-k)
$$

A beta Distribution with $\alpha$ and $\beta$ has the following mean and variance.
A beta distribution with $\alpha$ and $\beta$ has the following mean and variance.

The mean is $\frac{\alpha}{\alpha + \beta}$

Expand Down Expand Up @@ -679,4 +679,4 @@ Thus, the Bayesian statististian comes to believe that $\theta$ is near $.4$.

As shown in the figure above, as the number of observations grows, the Bayesian coverage intervals (BCIs) become narrower and narrower around $0.4$.

However, if you take a closer look, you will find that the centers of the are not exactly $0.4$, due to the persistent influence of the prior distribution and the randomness of the simulation path.
However, if you take a closer look, you will find that the centers of the BCIs are not exactly $0.4$, due to the persistent influence of the prior distribution and the randomness of the simulation path.