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Ines B.

Tutor for 2 years

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French

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Question:

Decris un ami proche en deux lignes. (physique, caractere, profession...)

Ines B.

Answer:

Mon ami s'appelle isabelle, elle est petite de taille, blonde aux yeux bleu. Isabelle est tres intelligente et elle est professeur par profession.

Linear Algebra

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Question:

Show the following: (a) If {v1, . . . , vn} is a spanning set for a vector space V and v is any vector in V , then v, v1, . . . , vn are linearly dependent. (b) If v1, . . . , vn are linearly independent vectors in a vector space V , then v2, . . . , vn cannot span V .

Ines B.

Answer:

Since {v1, . . . , vn} is a spanning set for V and v is in V , there are scalars α1, . . . , αn such that v = α1v1 + · · · + αnvn . Thus α1v1 + · · · + αnvn − v = 0 , so 0 can be written as a linear combination of v, v1, . . . , vn, with not all coefficients vanishing (the coefficient of v is −1 6= 0), so v, v1, . . . , vn are linearly dependent. (b) By contradiction. Let v1, . . . , vn be linearly independent. Suppose to the contrary that {v2, . . . , vn} is a spanning set for V . Since v1 ∈ V , by (a), the vectors v1, v2, . . . , vn must be linearly dependent, which is a contradiction. So the vectors v2, . . . , vn cannot span V

Statistics

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Question:

A sample of size n is taken from a N (µ, σ2 ) population, with µ known and σ 2 unknown. You wish to test null-hypothesis H0 : σ 2 = σ 2 0 against H1 : σ 2 6= σ 2 0 with significance level α. (a) Suggest a z-test (i.e. with normally distributed test statistic).

Ines B.

Answer:

(a) Under H0 i.e. when X1, . . . , Xn are iid N (µ, σ2 0 ) we have Z := X¯ − µ σ0/ √ n ∼ N (0, 1). The critical region for the 2-sided test is (−∞, −zα/2) ∪ (zα/2 ,∞).

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