math121b:final
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math121b:final [2020/05/04 19:29] pzhou [3. Probability and Statistics (20 pts)] |
math121b:final [2020/05/06 11:26] (current) pzhou [3. Probability and Statistics (20 pts)] |
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| ===== Final ===== | ===== Final ===== | ||
| $$\gdef\E{\mathbb E}$$ | $$\gdef\E{\mathbb E}$$ | ||
| + | ** Due Date **: May 10th (Sunday) 11:59PM. Submit online to gradescope. | ||
| + | ** Policy **: You can use textbook and your notes. There should be no discussion or collaborations, | ||
| + | ----- | ||
| ==== 1. Vector spaces and Curvilinear Coordinates (30 pts) ==== | ==== 1. Vector spaces and Curvilinear Coordinates (30 pts) ==== | ||
| All vectors spaces are finite dimensional over $\R$. | All vectors spaces are finite dimensional over $\R$. | ||
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| - | 2. (10 pt) Let $V_n$ be the vector space of polynomials whose degree is at most $n$. Let $f(x)$ be any smooth function on $[-1,1]$. Show that there is a unique element $f_n \in V_n$, such that for any $g \in V_n$, we have | + | 2. (10 pt) Let $V_n$ be the vector space of polynomials whose degree is at most $n$. Let $f(x)$ be any smooth function on $[-1,1]$. We fix $f(x)$ once and for all. Show that there is a unique element $f_n \in V_n$ (depending on our choice of $f$), such that for any $g \in V_n$, we have |
| $$ \int_{-1}^1 f_n(x) g(x) dx = | $$ \int_{-1}^1 f_n(x) g(x) dx = | ||
| - | Hint: (1) Equip $V_n$ with an inner product $\la g_1, g_2 \ra = \int_{-1}^1 g_1(x) g_2(x) dx$. (2) Show that $f(x)$ induces an element in $V_n^*$: $g \mapsto \int_{-1}^1 f(x) g(x) dx$. (3) use inner product to identify $V_n$ and $V_n^*$. | + | |
| + | //Hint: // | ||
| + | * (1) Equip $V_n$ with an inner product $\la g_1, g_2 \ra = \int_{-1}^1 g_1(x) g_2(x) dx$. | ||
| + | * (2) Show that $f(x)$ induces an element in $V_n^*$: $g \mapsto \int_{-1}^1 f(x) g(x) dx$. | ||
| + | * (3) use inner product to identify $V_n$ and $V_n^*$. | ||
| + | |||
| + | A remark: if $f(x)$ were a polynomial of degree less than $n$, then you could just take $f_n(x) = f(x)$. But, we have limited our choices of $f_n$ to be just degree $ \leq n$ polynomial, so we are looking for a 'best approximation' | ||
| 3. (10 pt) Let $\R^3$ be equipped with curvilinear coordinate $(u,v,w)$ where | 3. (10 pt) Let $\R^3$ be equipped with curvilinear coordinate $(u,v,w)$ where | ||
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| 4. (5 pt) Consider a random walk on the real line: at $t=0$, one start at $x=0$. Let $S_n$ denote the position at $t=n$, then $S_n = S_{n-1} + X_n$, where $X_n = \pm 1$ with equal probability. | 4. (5 pt) Consider a random walk on the real line: at $t=0$, one start at $x=0$. Let $S_n$ denote the position at $t=n$, then $S_n = S_{n-1} + X_n$, where $X_n = \pm 1$ with equal probability. | ||
| * (3pt) What is the variance of $S_n$? | * (3pt) What is the variance of $S_n$? | ||
| - | * (2pt) Use Markov inequality, prove that | + | * (2pt) Use Markov inequality, prove that for any $c > 1$, we have |
| $$ \P(|S_n| > c \sqrt{n}) \leq 1/c^2 $$ | $$ \P(|S_n| > c \sqrt{n}) \leq 1/c^2 $$ | ||
math121b/final.1588645742.txt.gz · Last modified: 2020/05/04 19:29 by pzhou
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