Choosing the right scale of initialization. In this exercise, you will analyze a single fully-connected layer where each weight is drawn from a Gaussian distribution. Your goal is to derive constraints on the variance of these weights so that the signal neither vanishes nor explodes as it propagates through layers. a. Consider the forward pass through a fully-connected linear layer with nin input inputs and nout outputs. Let the weights be initialized as N(0, 2 ) and assume that the inputs xi to the layer are independent, zero-mean random variables with variance var(xi) = v 2 . Biases are set to zero. If aj is the activation of the j th neuron, provide a step-by-step derivation to calculate var(aj ) in terms of 2 , v 2 , and nin.