About Posts which Tagged by 'Probability'
Let $X$ be a random variable. If $\mathscr{E}(X^2)=1$ and $\mathscr{E}|X|\geq a>0$, then $$\mathscr{P}\{|X|\geq\lambda a\}\geq(1-\lambda)^2a^2\mbox{ for }0\leq\lambda\leq1.$$
$\bullet$ Proof.
2015年9月1日 星期二
2015年8月26日 星期三
Proof of Chebyshev Type for Maximal Sum of Random Variables II
About Posts which Tagged by 'Probability'
Chebyshev type for maximal sum of random variables II. Let $\{X_n\}$ be independent random variables with finite means and suppose that there exists an $A$ such that $$\forall\,n,\,|X_n-\mathscr{E}(X_n)|\leq A<\infty,$$Then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$
See List of Inequalities.
$\bullet$ Proof.
Chebyshev type for maximal sum of random variables II. Let $\{X_n\}$ be independent random variables with finite means and suppose that there exists an $A$ such that $$\forall\,n,\,|X_n-\mathscr{E}(X_n)|\leq A<\infty,$$Then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$
See List of Inequalities.
$\bullet$ Proof.
Proof of Chebyshev Type for Maximal Sum of Random Variables I
Declaration for Posts which Tagged by 'Probability'
Chebyshev type for maximal sum of random variables I. Let $\{X_n\}$ be independent random variables such that $\mathscr{E}(X_n)=0$ and $\mathscr{E}(X_n^2)=\sigma^2(X_n)<\infty$ for all $n$, then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|>\varepsilon\right\}\leq\frac{\sigma^2(S_n)}{\varepsilon^2}.$$
See List of Inequalities.
$\bullet$ Proof.
Chebyshev type for maximal sum of random variables I. Let $\{X_n\}$ be independent random variables such that $\mathscr{E}(X_n)=0$ and $\mathscr{E}(X_n^2)=\sigma^2(X_n)<\infty$ for all $n$, then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|>\varepsilon\right\}\leq\frac{\sigma^2(S_n)}{\varepsilon^2}.$$
See List of Inequalities.
$\bullet$ Proof.
2015年8月24日 星期一
Proof of Cantelli's Inequality
About Posts which Tagged by 'Probability'
Cantelli's inequality. Suppose $\sigma^2=\mbox{Var}(X)<\infty$. Then for $a>0$, we have $$\mathscr{P}\{|X-\mathscr{E}(X)|>a\}\leq\frac{2\sigma^2}{a^2+\sigma^2}.$$
See List of Inequalities.
$\bullet$ Proof.
Cantelli's inequality. Suppose $\sigma^2=\mbox{Var}(X)<\infty$. Then for $a>0$, we have $$\mathscr{P}\{|X-\mathscr{E}(X)|>a\}\leq\frac{2\sigma^2}{a^2+\sigma^2}.$$
See List of Inequalities.
$\bullet$ Proof.
Proof of Inequality (6)
About Posts which Tagged by 'Probability'
Let $X$ and $Y$ be random variables. If $X\geq0$ and $Y\geq0$, $p\geq0$, then $$\mathscr{E}\{(X+Y)^p\}\leq2^p\{\mathscr{E}(X^p)+\mathscr{E}(Y^p)\}.$$If $p>1$, the factor $2^p$ may be replaced by $2^{p-1}$. If $0\leq p\leq1$, it may be replaced by $1$.
See List of Inequalities.
$\bullet$ Proof.
Let $X$ and $Y$ be random variables. If $X\geq0$ and $Y\geq0$, $p\geq0$, then $$\mathscr{E}\{(X+Y)^p\}\leq2^p\{\mathscr{E}(X^p)+\mathscr{E}(Y^p)\}.$$If $p>1$, the factor $2^p$ may be replaced by $2^{p-1}$. If $0\leq p\leq1$, it may be replaced by $1$.
See List of Inequalities.
$\bullet$ Proof.
2015年8月23日 星期日
Proof of Lyapunov's Inequality
About Posts which Tagged by 'Probability'
Lyapunov's inequality. Let $X$ be a random variable. For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$
See List of Inequalities.
$\bullet$ Proof.
Lyapunov's inequality. Let $X$ be a random variable. For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$
See List of Inequalities.
$\bullet$ Proof.
Proof of Chebyshev's inequality
About Posts which Tagged by 'Probability'
Chebyshev's inequality. Let $X$ be a random variable. Let $\phi$ be a strictly increasing function on $(0,\infty)$ and $\phi(u)=\phi(-u)$. Suppose $\mathscr{E}[\phi(X)]<\infty$. Then $\forall\,u>0$, $$\mathscr{P}\{|X|\geq u\}\leq\frac{\mathscr{E}[\phi(X)]}{\phi(u)}.$$
See List of Inequalities.
$\bullet$ Proof.
Chebyshev's inequality. Let $X$ be a random variable. Let $\phi$ be a strictly increasing function on $(0,\infty)$ and $\phi(u)=\phi(-u)$. Suppose $\mathscr{E}[\phi(X)]<\infty$. Then $\forall\,u>0$, $$\mathscr{P}\{|X|\geq u\}\leq\frac{\mathscr{E}[\phi(X)]}{\phi(u)}.$$
See List of Inequalities.
$\bullet$ Proof.
Proof of Jensen's inequality
About Posts which Tagged by 'Probability'
Jensen's inequality. Let $X$ be a random variable. Let $\phi$ be a convex function. Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$
See List of Inequalities.
$\bullet$ Proof.
Jensen's inequality. Let $X$ be a random variable. Let $\phi$ be a convex function. Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$
See List of Inequalities.
$\bullet$ Proof.
Proof of Minkowski's inequality
About Posts which Tagged by 'Probability'
Minkowski's inequality. Let $X$ and $Y$ are random variables. Let $1<p<\infty$. $$\left(\mathscr{E}|X+Y|^p\right)^{\frac{1}{p}}\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}+\left(\mathscr{E}|Y|^p\right)^{\frac{1}{p}}.$$
See List of Inequalities.
$\bullet$ Proof.
Minkowski's inequality. Let $X$ and $Y$ are random variables. Let $1<p<\infty$. $$\left(\mathscr{E}|X+Y|^p\right)^{\frac{1}{p}}\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}+\left(\mathscr{E}|Y|^p\right)^{\frac{1}{p}}.$$
See List of Inequalities.
$\bullet$ Proof.
Proof of H$\ddot{o}$lder's inequality
About Posts which Tagged by 'Probability'
H$\ddot{o}$lder's inequality. Let $X$ and $Y$ are random variables. Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. $$|\mathscr{E}(XY)|\leq \mathscr{E}|XY|\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}\left(\mathscr{E}|Y|^q\right)^{\frac{1}{q}}.$$
See List of Inequalities.
$\bullet$ Proof.
H$\ddot{o}$lder's inequality. Let $X$ and $Y$ are random variables. Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. $$|\mathscr{E}(XY)|\leq \mathscr{E}|XY|\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}\left(\mathscr{E}|Y|^q\right)^{\frac{1}{q}}.$$
See List of Inequalities.
$\bullet$ Proof.
2015年8月13日 星期四
Inequalities for Random Variable
About Posts which Tagged by 'Probability'
Let $X$ and $Y$ be random variables.
(1) [See Proof] H$\ddot{o}$lder's inequality. Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. $$|\mathscr{E}(XY)|\leq \mathscr{E}|XY|\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}\left(\mathscr{E}|Y|^q\right)^{\frac{1}{q}}.$$
(2) [See Proof] Minkowski's inequality. Let $1<p<\infty$. $$\left(\mathscr{E}|X+Y|^p\right)^{\frac{1}{p}}\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}+\left(\mathscr{E}|Y|^p\right)^{\frac{1}{p}}.$$
(3) [See Proof] Lyapunov's inequality. For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$
(4) [See Proof] Jensen's inequality. Let $\phi$ be a convex function. Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$
(5) [See Proof] Chebyshev's inequality. Let $\phi$ be a strictly increasing function on $(0,\infty)$ and $\phi(u)=\phi(-u)$. Suppose $\mathscr{E}[\phi(X)]<\infty$. Then $\forall\,u>0$, $$\mathscr{P}\{|X|\geq u\}\leq\frac{\mathscr{E}[\phi(X)]}{\phi(u)}.$$
(6) [See Proof] If $X\geq0$ and $Y\geq0$, $p\geq0$, then $$\mathscr{E}\{(X+Y)^p\}\leq2^p\{\mathscr{E}(X^p)+\mathscr{E}(Y^p)\}.$$If $p>1$, the factor $2^p$ may be replaced by $2^{p-1}$. If $0\leq p\leq1$, it may be replaced by $1$.
(7) [See Proof] Cantelli's inequality. Suppose $\sigma^2=\mbox{Var}(X)<\infty$. Then for $a>0$, we have $$\mathscr{P}\{|X-\mathscr{E}(X)|>a\}\leq\frac{2\sigma^2}{a^2+\sigma^2}.$$
(8) [See Proof] Chebyshev type for maximal sum of random variables I. Let $\{X_n\}$ be independent random variables such that $\mathscr{E}(X_n)=0$ and $\mathscr{E}(X_n^2)=\sigma^2(X_n)<\infty$ for all $n$, then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|>\varepsilon\right\}\leq\frac{\sigma^2(S_n)}{\varepsilon^2}.$$
(9) [See Proof] Chebyshev type for maximal sum of random variables II. Let $\{X_n\}$ be independent random variables with finite means and suppose that there exists an $A$ such that $$\forall\,n,\,|X_n-\mathscr{E}(X_n)|\leq A<\infty,$$Then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$
(10) [See Proof] If $\mathscr{E}(X^2)=1$ and $\mathscr{E}|X|\geq a>0$, then $$\mathscr{P}\{|X|\geq\lambda a\}\geq(1-\lambda)^2a^2\mbox{ for }0\leq\lambda\leq1.$$
Let $X$ and $Y$ be random variables.
(1) [See Proof] H$\ddot{o}$lder's inequality. Let $1<p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. $$|\mathscr{E}(XY)|\leq \mathscr{E}|XY|\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}\left(\mathscr{E}|Y|^q\right)^{\frac{1}{q}}.$$
(2) [See Proof] Minkowski's inequality. Let $1<p<\infty$. $$\left(\mathscr{E}|X+Y|^p\right)^{\frac{1}{p}}\leq \left(\mathscr{E}|X|^p\right)^{\frac{1}{p}}+\left(\mathscr{E}|Y|^p\right)^{\frac{1}{p}}.$$
(3) [See Proof] Lyapunov's inequality. For $0<s<t$, $$\left(\mathscr{E}|X|^s\right)^\frac{1}{s}\leq \left(\mathscr{E}|X|^t\right)^\frac{1}{t}.$$
(4) [See Proof] Jensen's inequality. Let $\phi$ be a convex function. Suppose $X$ and $\phi(X)$ are integrable. $$\phi(\mathscr{E}X)\leq \mathscr{E}[\phi(X)].$$
(5) [See Proof] Chebyshev's inequality. Let $\phi$ be a strictly increasing function on $(0,\infty)$ and $\phi(u)=\phi(-u)$. Suppose $\mathscr{E}[\phi(X)]<\infty$. Then $\forall\,u>0$, $$\mathscr{P}\{|X|\geq u\}\leq\frac{\mathscr{E}[\phi(X)]}{\phi(u)}.$$
(6) [See Proof] If $X\geq0$ and $Y\geq0$, $p\geq0$, then $$\mathscr{E}\{(X+Y)^p\}\leq2^p\{\mathscr{E}(X^p)+\mathscr{E}(Y^p)\}.$$If $p>1$, the factor $2^p$ may be replaced by $2^{p-1}$. If $0\leq p\leq1$, it may be replaced by $1$.
(7) [See Proof] Cantelli's inequality. Suppose $\sigma^2=\mbox{Var}(X)<\infty$. Then for $a>0$, we have $$\mathscr{P}\{|X-\mathscr{E}(X)|>a\}\leq\frac{2\sigma^2}{a^2+\sigma^2}.$$
(8) [See Proof] Chebyshev type for maximal sum of random variables I. Let $\{X_n\}$ be independent random variables such that $\mathscr{E}(X_n)=0$ and $\mathscr{E}(X_n^2)=\sigma^2(X_n)<\infty$ for all $n$, then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|>\varepsilon\right\}\leq\frac{\sigma^2(S_n)}{\varepsilon^2}.$$
(9) [See Proof] Chebyshev type for maximal sum of random variables II. Let $\{X_n\}$ be independent random variables with finite means and suppose that there exists an $A$ such that $$\forall\,n,\,|X_n-\mathscr{E}(X_n)|\leq A<\infty,$$Then let $S_n=\sum_{j=1}^nX_j$, we have for every $\varepsilon>0$, $$\mathscr{P}\left\{\underset{1\leq j\leq n}{\max}|S_j|\leq\varepsilon\right\}\leq\frac{(2A+4\varepsilon)^2}{\sigma^2(S_n)}.$$
(10) [See Proof] If $\mathscr{E}(X^2)=1$ and $\mathscr{E}|X|\geq a>0$, then $$\mathscr{P}\{|X|\geq\lambda a\}\geq(1-\lambda)^2a^2\mbox{ for }0\leq\lambda\leq1.$$
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