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Jiafeng L.

Graduate school student

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Mandarin

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Question:

write the Pinyin for the sentence below: 世界你好！ (Hello world!)

Jiafeng L.

Answer:

shi jie ni hao!

Linear Algebra

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Question:

Use Gauss-Jordan elimination to solve the system: x - 2y + 3z = 9 -x + 3y = -4 2x - 5y +5z =17

Jiafeng L.

Answer:

First use Gaussian elimination to obtain the row-echelon form: [1 -2 3 9; 0 1 3 5; 0 0 1 2] then apply elementary row operations: R1+2R2 -> R1: [1 0 9 19; 0 1 3 5; 0 0 1 2] R2-3R2 -> R2 [1 0 9 19; 0 1 0 -1; 0 0 1 2] R1-R3 -> R1: [1 0 0 1; 0 1 0 -1; 0 0 1 2] so finally x = 1, y = -1, z = 2.

Statistics

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Question:

There is an unbalanced coin. If you flip it you could get the probability of head P(head) = 0.6 and tail P(tail) = 0.4. Then how can we use it to predict an event that would happen with a probability of 0.5?

Jiafeng L.

Answer:

Because the coin is not balanced, you cannot get a 0.5 probability in only one flip. Then you could think about flip it twice, and there are 4 cases totally: {head,head},{head,tail},{tail,head},{tail,tail}. Calculate the joint probability of each case: P[head,head] = 0.6*0.6 = 0.36, P[head,tail] = P[tail,head] = 0.6 * 0.4 = 0.24, P[tail,tail] = 0.4 * 0.4 = 0.16. Since probabilities of case {head,tail}, {tail,head} are equal, we could only keep these two case to predict the given event and throw the other cases. So we could define case {head,tail} for the event would happen and {tail,head} for the event would not happen.

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