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 softmax$$(\boldsymbol{x})_{i} = {\frac {\text{exp}(x_{i})}{\sum _{j=1}^{n}\text{exp}(x_{j})}}$$

Softmax Function Definition

A model produces a vector of raw, unnormalized scores for three classes: `[3.0, 1.0, 0.2]`. To convert these scores into a probability distribution, a specific function is applied. This function first calculates the exponential of each score, and then normalizes these values by dividing each exponentiated score by the sum of all the exponentiated scores. Calculate the final probability value corresponding to the initial score of `3.0`. Show the main steps of your calculation and provide the final answer rounded to two decimal places.

Calculating a Probability Distribution

Consider a vector of scores **x** = [x₁, x₂, ..., xₙ]. A new vector **y** is created by adding a constant value C to every element of **x**, such that yᵢ = xᵢ + C for all i. How does the output of the softmax function for **y** compare to the output for **x**?

True or False: For any given non-empty vector of real numbers **x**, the sum of all the components of the resulting vector **y** = softmax(**x**) will always be equal to 1. The function is defined as: `softmax(x)_i = exp(x_i) / Σ exp(x_j)`.

In the context of converting a vector of raw numerical scores into a probability distribution, what is the primary role of the denominator (the summation term `Σ exp(x_j)`) in the softmax function `softmax(x)_i = exp(x_i) / Σ exp(x_j)`?

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