probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for...

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Random variables and probability mass/density functions 1 riv X S R probability mass function pint density disrete p fg Pr X x T R X f Ia pxC4 Px 2 o

Transcript of probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for...

Page 1: probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for X ~ Bin(100,0.5) k P(X=k) µ±σ binomial tails 3 For a random variable X, the tails

Random variables and probability mass/density functions

1

riv

X S R

probability mass function pintdensity

disrete p fg Pr X x

TR X f Ia

pxC4

Px 2 o

Page 2: probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for X ~ Bin(100,0.5) k P(X=k) µ±σ binomial tails 3 For a random variable X, the tails

30 40 50 60 70

0.00

0.02

0.04

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0.08

PMF for X ~ Bin(100,0.5)

k

P(X=k)

µ ± σ

binomial random variable

2

fjBmCnX counts theofH's inn

indy com tosses

0 eahgwjmm.g.PT s

p.m.f

ifor Binfloo 0.5

Is E H E kP

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30 40 50 60 70

0.00

0.02

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PMF for X ~ Bin(100,0.5)

k

P(X=k)

µ ± σ

binomial tails

3

For a random variable X, the tails of X are the parts of the PMF that are “far” from its mean.

g k

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tail bounds

For a random variable X, the tails of X are the parts of the PMF that are “far” from its mean.

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tail bounds

Often, we want to bound the probability that a random variable X is “extreme.”

Such a bound is called a “tail bound”.

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Markov’s inequality

Suppose we know that X is always non-negative.

Theorem: If X is a non-negative random variable, then for every α > 0, we have

Equivalently

P(X ≥ cE[X]) ≤ 1/c6

Page 7: probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for X ~ Bin(100,0.5) k P(X=k) µ±σ binomial tails 3 For a random variable X, the tails

Markov’s inequality

Theorem: If X is a non-negative random variable, then for every α > 0, we have

Proof:

E[X] = Σx x p(x)

= Σx<α x p(x) + Σx≥α x p(x)

≥ 0 + Σx≥ααp(x)

= αP(X ≥ α)7

(x ≥ 0; α ≤ x)

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Variance and Chebyshev’s inequality

If we know more about a random variable, we can often use that to get better tail bounds.

Suppose we also know the variance.

Standard deviation = square root of variance

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Var[Y ] = E[(Y � E(Y ))2]<latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit>

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30 40 50 60 70

0.00

0.02

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PMF for X ~ Bin(100,0.5)

k

P(X=k)

µ ± σ

binomial random variable

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Var[Y ] = E[(Y � E(Y ))2]<latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit>

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Page 10: probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for X ~ Bin(100,0.5) k P(X=k) µ±σ binomial tails 3 For a random variable X, the tails

Chebyshev’s inequalityIf we know more about a random variable, we can often use that to get better tail bounds.

Suppose we also know the variance.

Theorem: If Y is an arbitrary random variable with E[Y] = μ, then, for any α > 0,

10

T.toPr fly M E E ta

Page 11: probability pint density - courses.cs.washington.edu30 40 50 60 70 0.00 0.02 0.04 0.06 0.08 PMF for X ~ Bin(100,0.5) k P(X=k) µ±σ binomial tails 3 For a random variable X, the tails

Chebyshev’s inequality

Theorem: If Y is an arbitrary random variable with μ = E[Y], then, for any α > 0,

X is non-negative, so we can apply Markov’s inequality:

Proof:

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Important Facts about Variance

Linearity of expectation always holds, i.e.,

Linearity of variance holds only if the random variables are independent.

Also, if is a constant, then

12

Var[Y ] = E[(Y � E(Y ))2]<latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit><latexit sha1_base64="qWnADJ5Lh9AROejOQeh3s8O1N8g=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEitAvLTBF0IxSl4LKCvTEdSyZN29DMheSMWIau3Pgqblwo4tZncOfbmF4W2vpD4OM/53Byfi8SXIFlfRuppeWV1bX0emZjc2t7x9zdq6kwlpRVaShC2fCIYoIHrAocBGtEkhHfE6zuDa7G9fo9k4qHwS0MI+b6pBfwLqcEtNU2D1vAHiCpETlymi6+wGUn1zwp55r5/F3RbZtZq2BNhBfBnkEWzVRpm1+tTkhjnwVABVHKsa0I3IRI4FSwUaYVKxYROiA95mgMiM+Um0zOGOFj7XRwN5T6BYAn7u+JhPhKDX1Pd/oE+mq+Njb/qzkxdM/dhAdRDCyg00XdWGAI8TgT3OGSURBDDYRKrv+KaZ9IQkEnl9Eh2PMnL0KtWLA135xmS5ezONLoAB2hHLLRGSqha1RBVUTRI3pGr+jNeDJejHfjY9qaMmYz++iPjM8fIFqW8A==</latexit>

E(X1 +X2 + . . .+Xn) = E(X1) + E(X2) + . . .+ E(Xn)<latexit sha1_base64="OQFL3dOz51v2e9rAlnki0qHhCPM=">AAACKHicbVBdS8MwFE3n15xfVR99CQ5hQxjtEPRFHIrg4wS3FbZR0jTdwtK0JKkwxn6OL/4VX0QU2au/xLTrw9y8kHDuOeeS3OPFjEplWTOjsLa+sblV3C7t7O7tH5iHR20ZJQKTFo5YJBwPScIoJy1FFSNOLAgKPUY63ugu1TvPREga8Sc1jkk/RANOA4qR0pRr3txXHNeG59Bx6/ruMT9SMmt5FV7DTK3qPgX16qIjZXjVNctWzcoKrgI7B2WQV9M1P3p+hJOQcIUZkrJrW7HqT5BQFDMyLfUSSWKER2hAuhpyFBLZn2SLTuGZZnwYREIfrmDGLk5MUCjlOPS0M0RqKJe1lPxP6yYquOpPKI8TRTiePxQkDKoIpqlBnwqCFRtrgLCg+q8QD5FAWOlsSzoEe3nlVdCu12yNHy/Kjds8jiI4AaegAmxwCRrgATRBC2DwAt7AJ/gyXo1349uYza0FI585Bn/K+PkFSr+faA==</latexit><latexit sha1_base64="OQFL3dOz51v2e9rAlnki0qHhCPM=">AAACKHicbVBdS8MwFE3n15xfVR99CQ5hQxjtEPRFHIrg4wS3FbZR0jTdwtK0JKkwxn6OL/4VX0QU2au/xLTrw9y8kHDuOeeS3OPFjEplWTOjsLa+sblV3C7t7O7tH5iHR20ZJQKTFo5YJBwPScIoJy1FFSNOLAgKPUY63ugu1TvPREga8Sc1jkk/RANOA4qR0pRr3txXHNeG59Bx6/ruMT9SMmt5FV7DTK3qPgX16qIjZXjVNctWzcoKrgI7B2WQV9M1P3p+hJOQcIUZkrJrW7HqT5BQFDMyLfUSSWKER2hAuhpyFBLZn2SLTuGZZnwYREIfrmDGLk5MUCjlOPS0M0RqKJe1lPxP6yYquOpPKI8TRTiePxQkDKoIpqlBnwqCFRtrgLCg+q8QD5FAWOlsSzoEe3nlVdCu12yNHy/Kjds8jiI4AaegAmxwCRrgATRBC2DwAt7AJ/gyXo1349uYza0FI585Bn/K+PkFSr+faA==</latexit><latexit sha1_base64="OQFL3dOz51v2e9rAlnki0qHhCPM=">AAACKHicbVBdS8MwFE3n15xfVR99CQ5hQxjtEPRFHIrg4wS3FbZR0jTdwtK0JKkwxn6OL/4VX0QU2au/xLTrw9y8kHDuOeeS3OPFjEplWTOjsLa+sblV3C7t7O7tH5iHR20ZJQKTFo5YJBwPScIoJy1FFSNOLAgKPUY63ugu1TvPREga8Sc1jkk/RANOA4qR0pRr3txXHNeG59Bx6/ruMT9SMmt5FV7DTK3qPgX16qIjZXjVNctWzcoKrgI7B2WQV9M1P3p+hJOQcIUZkrJrW7HqT5BQFDMyLfUSSWKER2hAuhpyFBLZn2SLTuGZZnwYREIfrmDGLk5MUCjlOPS0M0RqKJe1lPxP6yYquOpPKI8TRTiePxQkDKoIpqlBnwqCFRtrgLCg+q8QD5FAWOlsSzoEe3nlVdCu12yNHy/Kjds8jiI4AaegAmxwCRrgATRBC2DwAt7AJ/gyXo1349uYza0FI585Bn/K+PkFSr+faA==</latexit><latexit sha1_base64="OQFL3dOz51v2e9rAlnki0qHhCPM=">AAACKHicbVBdS8MwFE3n15xfVR99CQ5hQxjtEPRFHIrg4wS3FbZR0jTdwtK0JKkwxn6OL/4VX0QU2au/xLTrw9y8kHDuOeeS3OPFjEplWTOjsLa+sblV3C7t7O7tH5iHR20ZJQKTFo5YJBwPScIoJy1FFSNOLAgKPUY63ugu1TvPREga8Sc1jkk/RANOA4qR0pRr3txXHNeG59Bx6/ruMT9SMmt5FV7DTK3qPgX16qIjZXjVNctWzcoKrgI7B2WQV9M1P3p+hJOQcIUZkrJrW7HqT5BQFDMyLfUSSWKER2hAuhpyFBLZn2SLTuGZZnwYREIfrmDGLk5MUCjlOPS0M0RqKJe1lPxP6yYquOpPKI8TRTiePxQkDKoIpqlBnwqCFRtrgLCg+q8QD5FAWOlsSzoEe3nlVdCu12yNHy/Kjds8jiI4AaegAmxwCRrgATRBC2DwAt7AJ/gyXo1349uYza0FI585Bn/K+PkFSr+faA==</latexit>

Var[X1 + . . .+Xn] = Var[X1] + . . .+Var[Xn]<latexit sha1_base64="fpaYldA10XrrV+NcP3zWfhI0ptk=">AAACNnicbZBLSwMxFIUz9VXra9Slm2ARBKHMiKAboejGjVDBPqAdhkyatqGZzJDcEcvQX+XG3+GuGxeKuPUnmLazcFovBD7OOSG5J4gF1+A4E6uwsrq2vlHcLG1t7+zu2fsHDR0lirI6jUSkWgHRTHDJ6sBBsFasGAkDwZrB8HbqN5+Y0jySjzCKmReSvuQ9TgkYybfvO8CeIW0QNW63fBfjM9wR3Qi0gZYvPXyN8wkvF8l50vPtslNxZoOXwc2gjLKp+fZbpxvRJGQSqCBat10nBi8lCjgVbFzqJJrFhA5Jn7UNShIy7aWztcf4xChd3IuUORLwTP17IyWh1qMwMMmQwEAvelPxP6+dQO/KS7mME2CSzh/qJQJDhKcd4i5XjIIYGSBUcfNXTAdEEQqm6ZIpwV1ceRka5xXX8MNFuXqT1VFER+gYnSIXXaIqukM1VEcUvaAJ+kCf1qv1bn1Z3/NowcruHKLcWD+/uXqqIg==</latexit><latexit sha1_base64="fpaYldA10XrrV+NcP3zWfhI0ptk=">AAACNnicbZBLSwMxFIUz9VXra9Slm2ARBKHMiKAboejGjVDBPqAdhkyatqGZzJDcEcvQX+XG3+GuGxeKuPUnmLazcFovBD7OOSG5J4gF1+A4E6uwsrq2vlHcLG1t7+zu2fsHDR0lirI6jUSkWgHRTHDJ6sBBsFasGAkDwZrB8HbqN5+Y0jySjzCKmReSvuQ9TgkYybfvO8CeIW0QNW63fBfjM9wR3Qi0gZYvPXyN8wkvF8l50vPtslNxZoOXwc2gjLKp+fZbpxvRJGQSqCBat10nBi8lCjgVbFzqJJrFhA5Jn7UNShIy7aWztcf4xChd3IuUORLwTP17IyWh1qMwMMmQwEAvelPxP6+dQO/KS7mME2CSzh/qJQJDhKcd4i5XjIIYGSBUcfNXTAdEEQqm6ZIpwV1ceRka5xXX8MNFuXqT1VFER+gYnSIXXaIqukM1VEcUvaAJ+kCf1qv1bn1Z3/NowcruHKLcWD+/uXqqIg==</latexit><latexit sha1_base64="fpaYldA10XrrV+NcP3zWfhI0ptk=">AAACNnicbZBLSwMxFIUz9VXra9Slm2ARBKHMiKAboejGjVDBPqAdhkyatqGZzJDcEcvQX+XG3+GuGxeKuPUnmLazcFovBD7OOSG5J4gF1+A4E6uwsrq2vlHcLG1t7+zu2fsHDR0lirI6jUSkWgHRTHDJ6sBBsFasGAkDwZrB8HbqN5+Y0jySjzCKmReSvuQ9TgkYybfvO8CeIW0QNW63fBfjM9wR3Qi0gZYvPXyN8wkvF8l50vPtslNxZoOXwc2gjLKp+fZbpxvRJGQSqCBat10nBi8lCjgVbFzqJJrFhA5Jn7UNShIy7aWztcf4xChd3IuUORLwTP17IyWh1qMwMMmQwEAvelPxP6+dQO/KS7mME2CSzh/qJQJDhKcd4i5XjIIYGSBUcfNXTAdEEQqm6ZIpwV1ceRka5xXX8MNFuXqT1VFER+gYnSIXXaIqukM1VEcUvaAJ+kCf1qv1bn1Z3/NowcruHKLcWD+/uXqqIg==</latexit><latexit sha1_base64="fpaYldA10XrrV+NcP3zWfhI0ptk=">AAACNnicbZBLSwMxFIUz9VXra9Slm2ARBKHMiKAboejGjVDBPqAdhkyatqGZzJDcEcvQX+XG3+GuGxeKuPUnmLazcFovBD7OOSG5J4gF1+A4E6uwsrq2vlHcLG1t7+zu2fsHDR0lirI6jUSkWgHRTHDJ6sBBsFasGAkDwZrB8HbqN5+Y0jySjzCKmReSvuQ9TgkYybfvO8CeIW0QNW63fBfjM9wR3Qi0gZYvPXyN8wkvF8l50vPtslNxZoOXwc2gjLKp+fZbpxvRJGQSqCBat10nBi8lCjgVbFzqJJrFhA5Jn7UNShIy7aWztcf4xChd3IuUORLwTP17IyWh1qMwMMmQwEAvelPxP6+dQO/KS7mME2CSzh/qJQJDhKcd4i5XjIIYGSBUcfNXTAdEEQqm6ZIpwV1ceRka5xXX8MNFuXqT1VFER+gYnSIXXaIqukM1VEcUvaAJ+kCf1qv1bn1Z3/NowcruHKLcWD+/uXqqIg==</latexit>

Var[a ·X] = a2 ·Var[X]<latexit sha1_base64="8YWzrdKn+MiazbfULtdC8tfLQwc=">AAACFXicbZDLSgMxFIYzXmu9jbp0EyyCCykzRdCNUHTjsoK9wHQsZ9K0Dc1cSM6IZehLuPFV3LhQxK3gzrcxvSxq64HAl/8/h+T8QSKFRsf5sZaWV1bX1nMb+c2t7Z1de2+/puNUMV5lsYxVIwDNpYh4FQVK3kgUhzCQvB70r0d+/YErLeLoDgcJ90PoRqIjGKCRWvZpE/kjZjVQQw+arB0jbfj0ksJ9aXybsRt+yy44RWdcdBHcKRTItCot+7vZjlka8giZBK0910nQz0ChYJIP881U8wRYH7rcMxhByLWfjbca0mOjtGknVuZESMfq7EQGodaDMDCdIWBPz3sj8T/PS7Fz4WciSlLkEZs81EklxZiOIqJtoThDOTAATAnzV8p6oIChCTJvQnDnV16EWqnoGr49K5SvpnHkyCE5IifEJeekTG5IhVQJI0/khbyRd+vZerU+rM9J65I1nTkgf8r6+gV95Z8C</latexit><latexit sha1_base64="8YWzrdKn+MiazbfULtdC8tfLQwc=">AAACFXicbZDLSgMxFIYzXmu9jbp0EyyCCykzRdCNUHTjsoK9wHQsZ9K0Dc1cSM6IZehLuPFV3LhQxK3gzrcxvSxq64HAl/8/h+T8QSKFRsf5sZaWV1bX1nMb+c2t7Z1de2+/puNUMV5lsYxVIwDNpYh4FQVK3kgUhzCQvB70r0d+/YErLeLoDgcJ90PoRqIjGKCRWvZpE/kjZjVQQw+arB0jbfj0ksJ9aXybsRt+yy44RWdcdBHcKRTItCot+7vZjlka8giZBK0910nQz0ChYJIP881U8wRYH7rcMxhByLWfjbca0mOjtGknVuZESMfq7EQGodaDMDCdIWBPz3sj8T/PS7Fz4WciSlLkEZs81EklxZiOIqJtoThDOTAATAnzV8p6oIChCTJvQnDnV16EWqnoGr49K5SvpnHkyCE5IifEJeekTG5IhVQJI0/khbyRd+vZerU+rM9J65I1nTkgf8r6+gV95Z8C</latexit><latexit sha1_base64="8YWzrdKn+MiazbfULtdC8tfLQwc=">AAACFXicbZDLSgMxFIYzXmu9jbp0EyyCCykzRdCNUHTjsoK9wHQsZ9K0Dc1cSM6IZehLuPFV3LhQxK3gzrcxvSxq64HAl/8/h+T8QSKFRsf5sZaWV1bX1nMb+c2t7Z1de2+/puNUMV5lsYxVIwDNpYh4FQVK3kgUhzCQvB70r0d+/YErLeLoDgcJ90PoRqIjGKCRWvZpE/kjZjVQQw+arB0jbfj0ksJ9aXybsRt+yy44RWdcdBHcKRTItCot+7vZjlka8giZBK0910nQz0ChYJIP881U8wRYH7rcMxhByLWfjbca0mOjtGknVuZESMfq7EQGodaDMDCdIWBPz3sj8T/PS7Fz4WciSlLkEZs81EklxZiOIqJtoThDOTAATAnzV8p6oIChCTJvQnDnV16EWqnoGr49K5SvpnHkyCE5IifEJeekTG5IhVQJI0/khbyRd+vZerU+rM9J65I1nTkgf8r6+gV95Z8C</latexit><latexit sha1_base64="8YWzrdKn+MiazbfULtdC8tfLQwc=">AAACFXicbZDLSgMxFIYzXmu9jbp0EyyCCykzRdCNUHTjsoK9wHQsZ9K0Dc1cSM6IZehLuPFV3LhQxK3gzrcxvSxq64HAl/8/h+T8QSKFRsf5sZaWV1bX1nMb+c2t7Z1de2+/puNUMV5lsYxVIwDNpYh4FQVK3kgUhzCQvB70r0d+/YErLeLoDgcJ90PoRqIjGKCRWvZpE/kjZjVQQw+arB0jbfj0ksJ9aXybsRt+yy44RWdcdBHcKRTItCot+7vZjlka8giZBK0910nQz0ChYJIP881U8wRYH7rcMxhByLWfjbca0mOjtGknVuZESMfq7EQGodaDMDCdIWBPz3sj8T/PS7Fz4WciSlLkEZs81EklxZiOIqJtoThDOTAATAnzV8p6oIChCTJvQnDnV16EWqnoGr49K5SvpnHkyCE5IifEJeekTG5IhVQJI0/khbyRd+vZerU+rM9J65I1nTkgf8r6+gV95Z8C</latexit>

a<latexit sha1_base64="DcxH1t8VfbGREJT8ZMrfQnx14cc=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEEiuDau++0UNja3tneKu6W9/YPDo/LxSVvHqWLYYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFFjrSYdlCtu1V2IrIOXQwVyNQblr/4wZmmE0jBBte55bmL8jCrDmcBZqZ9qTCib0BH2LEoaofazxaIzcmGdIQljZZ80ZOH+nshopPU0CmxnRM1Yr9bm5n+1XmrCGz/jMkkNSrb8KEwFMTGZX02GXCEzYmqBMsXtroSNqaLM2GxKNgRv9eR1aF9VPcvN60r9No+jCGdwDpfgQQ3qcA8NaAEDhGd4hTfn0Xlx3p2PZWvByWdO4Y+czx/DX4zl</latexit><latexit sha1_base64="DcxH1t8VfbGREJT8ZMrfQnx14cc=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEEiuDau++0UNja3tneKu6W9/YPDo/LxSVvHqWLYYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFFjrSYdlCtu1V2IrIOXQwVyNQblr/4wZmmE0jBBte55bmL8jCrDmcBZqZ9qTCib0BH2LEoaofazxaIzcmGdIQljZZ80ZOH+nshopPU0CmxnRM1Yr9bm5n+1XmrCGz/jMkkNSrb8KEwFMTGZX02GXCEzYmqBMsXtroSNqaLM2GxKNgRv9eR1aF9VPcvN60r9No+jCGdwDpfgQQ3qcA8NaAEDhGd4hTfn0Xlx3p2PZWvByWdO4Y+czx/DX4zl</latexit><latexit sha1_base64="DcxH1t8VfbGREJT8ZMrfQnx14cc=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEEiuDau++0UNja3tneKu6W9/YPDo/LxSVvHqWLYYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFFjrSYdlCtu1V2IrIOXQwVyNQblr/4wZmmE0jBBte55bmL8jCrDmcBZqZ9qTCib0BH2LEoaofazxaIzcmGdIQljZZ80ZOH+nshopPU0CmxnRM1Yr9bm5n+1XmrCGz/jMkkNSrb8KEwFMTGZX02GXCEzYmqBMsXtroSNqaLM2GxKNgRv9eR1aF9VPcvN60r9No+jCGdwDpfgQQ3qcA8NaAEDhGd4hTfn0Xlx3p2PZWvByWdO4Y+czx/DX4zl</latexit><latexit sha1_base64="DcxH1t8VfbGREJT8ZMrfQnx14cc=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeqx6MVjC/YD2lA220m7drMJuxuhhP4CLx4U8epP8ua/cdvmoK0vLDy8M8POvEEiuDau++0UNja3tneKu6W9/YPDo/LxSVvHqWLYYrGIVTegGgWX2DLcCOwmCmkUCOwEk7t5vfOESvNYPphpgn5ER5KHnFFjrSYdlCtu1V2IrIOXQwVyNQblr/4wZmmE0jBBte55bmL8jCrDmcBZqZ9qTCib0BH2LEoaofazxaIzcmGdIQljZZ80ZOH+nshopPU0CmxnRM1Yr9bm5n+1XmrCGz/jMkkNSrb8KEwFMTGZX02GXCEzYmqBMsXtroSNqaLM2GxKNgRv9eR1aF9VPcvN60r9No+jCGdwDpfgQQ3qcA8NaAEDhGd4hTfn0Xlx3p2PZWvByWdO4Y+czx/DX4zl</latexit>

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Performance/estimation using repetition

13


High-resolution Search for Q+ Pentaquark in p–p K–X Reaction

High-resolution Search for Q+ Pentaquark in p–p K–X Reaction

52 Das Newton-Verfahren...52 Das Newton-Verfahren 52.1 Motivation Das Konvergenzverhalten von Fixpunkt-Iterationen x k+1 = Φ(x k), k ∈N zur L¨osung nichtlinearer Gleichungen h

52 Das Newton-Verfahren...52 Das Newton-Verfahren 52.1 Motivation Das Konvergenzverhalten von Fixpunkt-Iterationen x k+1 = Φ(x k), k ∈N zur L¨osung nichtlinearer Gleichungen h

Study of B X(3872) K π - TIFRgmohanty/baw2012/anu1.pdf2 Motivation ⧫ X(3872) not a conventional charmonium state ⧫ Study of B X3872 K π (by K. Trabelsi, BN 999) B X3872 K π

Study of B X(3872) K π - TIFRgmohanty/baw2012/anu1.pdf2 Motivation ⧫ X(3872) not a conventional charmonium state ⧫ Study of B X3872 K π (by K. Trabelsi, BN 999) B X3872 K π

1. CIRCULAR FUNCTIONS 1. The cotangent as an infinite ...vsv/fullnotes.pdf · X∞ n=−∞ 1 (z+ n)k (k≥1) (3) For k≥2 the series converges absolutely, but for k= 1 we have

1. CIRCULAR FUNCTIONS 1. The cotangent as an infinite ...vsv/fullnotes.pdf · X∞ n=−∞ 1 (z+ n)k (k≥1) (3) For k≥2 the series converges absolutely, but for k= 1 we have

2D Convolution/Multiplication Application of Convolution Thm. · 2015. 10. 19. · Convolution F[g(x,y)**h(x,y)]=G(k x,k y)H(k x,k y) Multiplication F[g(x,y)h(x,y)]=G(k x,k y)**H(k

2D Convolution/Multiplication Application of Convolution Thm. · 2015. 10. 19. · Convolution F[g(x,y)**h(x,y)]=G(k x,k y)H(k x,k y) Multiplication F[g(x,y)h(x,y)]=G(k x,k y)**H(k

PK T yTg*sin PK - geoservis.ftn.uns.ac.rs i 9 Vezba.pdf · 10 Računanje koeficijenta deformacije papira m y t Y k Y = m x t X k X = Y t i X t su teoretske vrednosti 50 mm. X m i

PK T yTg*sin PK - geoservis.ftn.uns.ac.rs i 9 Vezba.pdf · 10 Računanje koeficijenta deformacije papira m y t Y k Y = m x t X k X = Y t i X t su teoretske vrednosti 50 mm. X m i

Conduction Heat Transfer - UniMAP Portalportal.unimap.edu.my/portal/page/portal30/Lecturer Notes... · composite wall c c B B A A x T T k A x T T k A x T T Q k A Δ − = − Δ −

Conduction Heat Transfer - UniMAP Portalportal.unimap.edu.my/portal/page/portal30/Lecturer Notes... · composite wall c c B B A A x T T k A x T T k A x T T Q k A Δ − = − Δ −

Universidad Autónoma Metropolitana - Iztapalapa División ... · PDF filePara la ecuación diferencial () K x f x dx d x − φ = ... transformada de Fourier puede escribirse de la

Universidad Autónoma Metropolitana - Iztapalapa División ... · PDF filePara la ecuación diferencial () K x f x dx d x − φ = ... transformada de Fourier puede escribirse de la

Octogon Mathematical Magazine, Vol. 23, No.1, April 2015 … · 2016-05-29 · x,y∈ R for which 2 n k=1 z x+y 2 k − n k=1 w x+y 2 k ≤ n k=1 zx k −w y k + n k=1 zy k −w x

Octogon Mathematical Magazine, Vol. 23, No.1, April 2015 … · 2016-05-29 · x,y∈ R for which 2 n k=1 z x+y 2 k − n k=1 w x+y 2 k ≤ n k=1 zx k −w y k + n k=1 zy k −w x

Rrs(λ - IOCCGRrs Rrs K d v Or, K Chl d (O) v. 2 3 b 1 a 1 2 ( 490 ) ( 490 ) a a K b X or K X d d (551 ) (488 ) ... With IOPs as inputs, many products, e.g. Kd, Zeu, Zsd, PP, could

Rrs(λ - IOCCGRrs Rrs K d v Or, K Chl d (O) v. 2 3 b 1 a 1 2 ( 490 ) ( 490 ) a a K b X or K X d d (551 ) (488 ) ... With IOPs as inputs, many products, e.g. Kd, Zeu, Zsd, PP, could

MATH 4211/6211 – Optimization Gradient MethodConsider x(k) and compute g(k):= rf(x(k)).Set descent direction to d(k) = g(k). Now we want to find 0 such that x(k) g(k) improves x(k).

MATH 4211/6211 – Optimization Gradient MethodConsider x(k) and compute g(k):= rf(x(k)).Set descent direction to d(k) = g(k). Now we want to find 0 such that x(k) g(k) improves x(k).

I 3 P06 GR14 - YPEKA › egyFloods › gr14 › reports › I_3_P06... · 2018-12-13 · - v z a g m g = v b \ o g p d q % g k a ` k x k + i d j j ` o = q r x k & b h = k c k n m

I 3 P06 GR14 - YPEKA › egyFloods › gr14 › reports › I_3_P06... · 2018-12-13 · - v z a g m g = v b \ o g p d q % g k a ` k x k + i d j j ` o = q r x k & b h = k c k n m

: g e b n d ` f ^ o d i b j Z h f j d g ^ f o d a f ^ g x ^ i n d o t i b ......¨ f b x e p i n d Q o l ^ o d ` f g k x Q r b a f ^ n k x R [ ^ N f g k i k f g y i K b h b o y i g

: g e b n d ` f ^ o d i b j Z h f j d g ^ f o d a f ^ g x ^ i n d o t i b ......¨ f b x e p i n d Q o l ^ o d ` f g k x Q r b a f ^ n k x R [ ^ N f g k i k f g y i K b h b o y i g

Length x Width 1200 mm x 600 mm fileTemperature I SC α [%/K] +0.02 Temperature V OC β [%/K] -0.24 Temperature P MPP γ [%/K] -0.25 under STC. IMPP, VMPP, ISC, VOC are within ±10%

Length x Width 1200 mm x 600 mm fileTemperature I SC α [%/K] +0.02 Temperature V OC β [%/K] -0.24 Temperature P MPP γ [%/K] -0.25 under STC. IMPP, VMPP, ISC, VOC are within ±10%

TRIGONOMETRI (Kompetensi 4) · Perkalian jumlah/selisih ... (x - α) dengan k = a2 +b2: persamaan lengkapnya: a cos x + b sin x = k cos (x - α) ... y = sin x a. Mempunyai nilai ...

TRIGONOMETRI (Kompetensi 4) · Perkalian jumlah/selisih ... (x - α) dengan k = a2 +b2: persamaan lengkapnya: a cos x + b sin x = k cos (x - α) ... y = sin x a. Mempunyai nilai ...

Point clustering with a convex, corrected K-means · k: X a= k+ E a with E[X a] = k and E a ˘ ind sub-N(0; a) Key quantities to keep in mind: cluster separation G( ) := min k6=lj

Point clustering with a convex, corrected K-means · k: X a= k+ E a with E[X a] = k and E a ˘ ind sub-N(0; a) Key quantities to keep in mind: cluster separation G( ) := min k6=lj

with: r) . ofstad - staff.uni-mainz.de fileof rm s ˆτ n (k = X x ∈ Z d τ n (x) e ik · x k ∈ [− π] d. r p c, ˆτ n (0) is small. r p c, ˆτ n (0) s n d. r p = p c, iour

with: r) . ofstad - staff.uni-mainz.de fileof rm s ˆτ n (k = X x ∈ Z d τ n (x) e ik · x k ∈ [− π] d. r p c, ˆτ n (0) is small. r p c, ˆτ n (0) s n d. r p = p c, iour

Dimensions and spectral triples - uniroma2.itisola/research/preprints/GuIs10.pdf · there are λj ∈ (0,1) such that kwj(x) − wj(y)k = λjkx − yk, x,y ∈ RN. Denote by K(RN)

Dimensions and spectral triples - uniroma2.itisola/research/preprints/GuIs10.pdf · there are λj ∈ (0,1) such that kwj(x) − wj(y)k = λjkx − yk, x,y ∈ RN. Denote by K(RN)

MA1101 MATEMATIKA 1A - WordPress.com · 1.3 Teorema-Teorema Limit Menggunakan teorema-teorema limit dalam ... Hendra Gunawan 10 lim kf ( x) k lim f ( x). x o c x o c lim k k. x c

MA1101 MATEMATIKA 1A - WordPress.com · 1.3 Teorema-Teorema Limit Menggunakan teorema-teorema limit dalam ... Hendra Gunawan 10 lim kf ( x) k lim f ( x). x o c x o c lim k k. x c

X-ray diffraction Biophysics. W. K. Röntgen (1845 – 1923) Wife’s hand.

X-ray diffraction Biophysics. W. K. Röntgen (1845 – 1923) Wife’s hand.