I believe the proposition that entropy cannot be negative applies only to categorical distributions, which are discrete distributions for which the categories have no intrinsic numerical values. For example, the outcome of a (stylized) coin flip can be characterized by two categories: Heads and Tails. One can associate the numbers 1 and 0 with those categories, but they are for convenience only. For example, with those numerical values the expectation of the random variable equals the probability of Heads. It is, however, the probability of Heads that is fundamental, not the expectation.

In the example given in the question, the distribution is continuous. Nevertheless, we can treat is as a categorical distribution by treating the bins as categories for which the bin centers (for example) are the category labels, which have no numerical significance. In this case, the entropy is given by \begin{equation} -\sum_i p_i\,\log(p_i) . \end{equation}

In Mathematica, one can do the following:

probs = Cases[p1, x_ * Boole[_] :> x/10];
-probs.Log[probs]

which produces 1.94194.

I believe the proposition that entropy cannot be negative applies only to categorical distributions, which are discrete distributions for which the categories have no intrinsic numerical values. For example, the outcome of a (stylized) coin flip can be characterized by two categories: Heads and Tails. One can associate the numbers 1 and 0 with those categories, but they are for convenience only. For example, with those numerical values the expectation of the random variable equals the probability of Heads. It is, however, the probability of Heads that is fundamental, not the expectation.

In the example given in the question, the distribution is continuous. Nevertheless, we can use the bin probabilities themselves --- purely as an example. We can obtain a categorical distribution by treating the bins as categories for which the bin centers (for example) are the category labels, which have no numerical significance. In this case, the entropy is given by \begin{equation} -\sum_i p_i\,\log(p_i) . \end{equation}

In Mathematica, one can do the following:

probs = Cases[p1, x_ * Boole[_] :> x/10];
-probs.Log[probs]

which produces 1.94194.

I believe the proposition that entropy cannot be negative applies only to categorical distributions, which are discrete distributions for which the categories have no intrinsic numerical values. For example, the outcome of a (styled) coin flip can be characterized by two categories: Heads and Tails. One can associate the numbers 1 and 0 with those categories, but they are for convenience only. For example, with those numerical values the expectation of the random variable equals the probability of Heads. It is, however, the probability of Heads that is fundamental, not the expectation.

In the example given in the question, the distribution is continuous. Nevertheless, we can treat is as a categorical distribution by treating the bins as categories for which the bin centers (for example) are the category labels, which have no numerical significance. In this case, the entropy is given by \begin{equation} -\sum_i p_i\,\log(p_i) . \end{equation}

In Mathematica, one can do the following:

probs = Cases[p1, x_ * Boole[_] :> x/10];
-probs.Log[probs]

which produces 1.94194.

I believe the proposition that entropy cannot be negative applies only to categorical distributions, which are discrete distributions for which the categories have no intrinsic numerical values. For example, the outcome of a (stylized) coin flip can be characterized by two categories: Heads and Tails. One can associate the numbers 1 and 0 with those categories, but they are for convenience only. For example, with those numerical values the expectation of the random variable equals the probability of Heads. It is, however, the probability of Heads that is fundamental, not the expectation.

In the example given in the question, the distribution is continuous. Nevertheless, we can treat is as a categorical distribution by treating the bins as categories for which the bin centers (for example) are the category labels, which have no numerical significance. In this case, the entropy is given by \begin{equation} -\sum_i p_i\,\log(p_i) . \end{equation}

In Mathematica, one can do the following:

probs = Cases[p1, x_ * Boole[_] :> x/10];
-probs.Log[probs]

which produces 1.94194.