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--- math121b:final [2020/05/04 01:46]
pzhou created
+++ math121b:final [2020/05/06 11:26] (current)
pzhou [3. Probability and Statistics (20 pts)]
@@ Line 1: / Line 1: @@
 ===== Final =====
  $\gdef\E{\mathbb E}$
+** 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, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email.
+-----
 ==== 1. Vector spaces and Curvilinear Coordinates (30 pts) ====
 All vectors spaces are finite dimensional over  $\R$ .
@@ Line 19: / Line 22: @@
-. (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
+. (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(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' of  $f(x)$  in  $V_n$  in a sense. Try solve the example case of  $n=1$ ,  $f(x) = \sin(x)$  if you need some intuition.
 . (10 pt) Let  $\R^3$  be equipped with curvilinear coordinate  $(u,v,w)$  where
@@ Line 70: / Line 79: @@
 Use the moment generating function to find out  $\E(X^4)$ . Let  $Y = X^2$ . What is the mean and variance of  $Y$ ?
-. (5 pt) There are two bags of balls. Bag A contains 4 black balls and 6 white balls, Bag B contains 10 black balls and 10 white balls. Suppose we randomly pick a bag and randomly pick a ball. Given that the ball is white, what is the probability that we picked bag A?
+. (5 pt) There are two bags of balls. Bag A contains 4 black balls and 6 white balls, Bag B contains 10 black balls and 10 white balls. Suppose we randomly pick a bag (with equal probability) and randomly pick a ball. Given that the ball is white, what is the probability that we picked bag A?
 . (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  $S_n$ ?
-  * (2pt) Use Markov inequality, prove that
+  * (2pt) Use Markov inequality, prove that for any  $c > 1$ , we have
     $\P(|S_n| > c \sqrt{n}) \leq 1/c^2$

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