math121b:final
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math121b:final [2020/05/05 17:07] pzhou |
math121b:final [2020/05/06 11:26] (current) pzhou [3. Probability and Statistics (20 pts)] |
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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.1588723647.txt.gz · Last modified: 2020/05/05 17:07 by pzhou
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