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In this problem, we're given a random variable X which has a uniform distribution in the interval negative 1 to 1. In other words, if we were to draw out the PDF of X, we see that in the interval negative 1 to 1, it has value 1/2. Now we're given a sequence random variables X1, X2, and so on, where each Xi has the same distribution as X and different Xi's are independent. For part a, we would like to know if the sequence Xi converges to some number-- let's call it c-- in probability as i goes to infinity-- whether this is true. Let's first recall the definition of convergence in probability. If this does happen, then by definition, we'll have that for every epsilon greater than 0, the probability Xi minus c greater equal to epsilon, this quantity will go to 0 in the limit of i going to infinity. In other words, with very high probability, we will find Xi to be very concentrated around the number c if this were to be the PDF of Xi. Now, can this...

We are given a CBit module shown here. This module takes corresponding bits of two unsigned binary numbers, A and B, along with two Cin bits from higher-order CBit modules. Its output bit R, is the appropriate bit of the larger among A and B. It also has two additional output bits Cout which indicate respectively if A>B or B>A in the bits considered so far. The propagation delay of each Cbit module is 4ns. The CBit module is used to create a product MAXC which is a combinational device that determines the maximum of its two 4-bit unsigned binary inputs. It is constructed using 4 CBit modules as shown here. The first question we want to consider is what are the propagation delay and throughput of this combinational circuit? The propagation delay of a combinational circuit is the propagation delay along its longest path from input to output. In this case, the longest path is through 4 CBit modules. Since each CBit module has a propagation delay of 4 ns, the propagatio...

We can consider a given coordinate system as a reference frame within which we can describe the kinematics of an object. By "the kinematics," I mean the position, the velocity, and the acceleration as a function of time, basically a geometric description of the motion. Some aspects of these kinematics will look different in different reference frames and I'd like to examine that now. First, I want to define what I mean by an "inertial reference frame." An inertial reference frame is one in which an isolated body, one with no net force acting on it, moves at constant velocity, where that constant velocity might be zero. Another way of saying this is that an inertial reference frame is one in which Newton's laws of motion apply. Recall that Newton's first law of motion states that an isolated object with no forces acting on it moves at constant velocity. So let's begin by considering an observer in a particular reference frame. We'll ...