Respond to two colleagues by sharing insights gained from their post and explain how you will integrate those insights into your own biopsychosocial assessment. You may also provid

3. Now we have our generators working, we must construct random stimuli. For our purposes, this means constructing vectors with random values. It is usually convenient to construct stimuli with no ‘DC’ component, that is, the distribution should have mean zero. This means that we should subtract the mean of the vector elements from the element values, or, very slightly differently, doing this probabilistically by subtracting 0.5 from the values generated by Random or your own computer’s function giving uniformly distributed random values. In practice, for reasonably large dimensionality, these techniques do not differ in their effects on the neural models. Your specific assignment will now be to study the behavior of vectors containing random elements. I would like you to generate normalized (i.e., having length 1) random vectors whose elements are taken from a distribution with mean zero. It is probably easiest to use the uniform distribution to generate the elements. Generate many pairs of such vectors and generate their dot [inner] products. (a) (1 point) What does this dot product actually mean, geometrically? (Remember, the length of the two vectors is 1.) (b) (4 points) Generate a histogram of dot products and compute mean and standard deviation of the dot products. Use several dimensionalities: 10, 20, 50, 100 250, 500, 1000 and 2000.) It is trivial to compute what the mean of the resulting distribution of dot products should be, given the constraints on the vectors. (c) (2 points) Tell me what it should be (and why) and compare it with your results. (d) (8 points) Computing the expected standard deviation of the distribution of dot products between these vectors is not so easy but not hard if you know statistics. If you can figure it out mathematically, tell me and compare it with your simulation. (5 points for 2-dimensional and 3- dimensional cases, 3 points for all dimensions beyond. Hint: Start from the definition of variance: Var(X) = E[X^2] – E[X]^2. Proceed by applying properties of expected values and what you know about how the vectors were generated.) (e) (4 points) Otherwise, see if you can figure out (i.e., guess) roughly what it should be from your ‘data’. Try to guess how the “width” (standard deviation) of the distribution changes with the dimensionality of the vectors used in the dot product. We would expect the width to decrease as the number of elements increases (wouldn’t we?). Some of this material is covered in the textbook. Reminder: please choose either (d) or (e), but not both. PLEASE INCLUDE PYTHON CODE