MAT 480 Topology Problem Set 1 Due 2/1/2019 Spring 2019 1. Let a (x1, z2, 23) and y - (1, 2, va) be elements of R3. Define d(x,y) = maxlx1-vil, la 2-U21, las-val). Prove that d defines a metric on R3 2. Let u(u1, u2, ..., un) and - (v, V2, ..., Vn) be vectors in Rn. Prove the Cauchy-Schwarz Inequality lin 히 lizil . Ilull in six easy steps: (a) Explain why lu.卵く间2·间2 implies lu. 히 (b) Explain why, for any real z, 阅·间. (c) Rewrite the right-hand side of the inequality above as a quadratic polynomial az2 + baz + c. What are the values of the coefficients a, b, and c? (d) Let A-b2 4ac. Explain why A s 0. Hint: (b) implies 0 az2 ba + c; what do you know about the graph of such a quadratic polynomial'? (e) Use (d) to show lž.2I2. l12 (f) Conclude that the Cauchy-Schwarz Inequality is true. 3. Given a metric space (X, d), a point a E X, and some e> 0, we can define the "open" e-ball centered at as Similarly, the "closed" e-ball is defined as B(r, e)(v E 지 d(x, y) e). (a) Let de be the Euclidean metric on R2. Sketch a picture of B(0,0), 1) in (R2, dB) (b) Let dr be the Taxi-cab metric on IR2. Sketch a picture of B((0,0), 1) in (R2, dr). (c) True or false, with explanation: given B((0,0),1) in (R2, dE) and any point V E B((0,0), 1), there is always some 6> 0 so that (d) Do we have the same answer if we replace B(0, 0), 1) with 瓦(0,0), 1) in (c)? Explain.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
I am taking a Topology course and was wondering if you could help me with Problem 2 and 3 of the worksheet that is attached.
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