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Medium 9780596514860

3. Rules for Numeric Operations: Follow the rules

Tracey Pilone O'Reilly Media ePub

Sometimes you just gotta follow the stinking rules.

Yes, there comes a time in everyone’s life when mom and dad are out of the picture, and there’s nobody making you clean your room or surrender your cell phone until your homework’s done. But when it comes to Algebra, rules are a good thing. They’ll keep you from getting the wrong answer. In fact, lots of times, rules will help you solve for an unknown without a lot of extra work. Leave your dunce cap behind for this chapter because we’ll be following a few handy rules all the way to a perfect score.

It’s the quiz show that’s sweeping the nation—Math or No Math. This new primetime hit pits two contestants against each other, struggling to solve math problems. It’s easy to find contestant, but Math or No Math needs help. They don’t have any judges to figure out if contestants are getting the problems right! That’s where you come in....

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Medium 9781576337370

Logarithms: CLEP Algebra II

Ace Academics Ace Academics ePub
Medium 9781576336830

Multiplication and Division: Decimals: Praxis I Arithmetic

Ace Academics Ace Academics ePub
Medium 9789380298818

14

Parmananda Gupta Laxmi Publications PDF

160

DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

∴ The extremals of the functional J[y(x)] can be found by solving Euler’s equation of the transformed functional J1[v(u)]. This is called the principle of invariance of Euler’s equation under coordinates transformations.

In solving practical problems, the change of variables is made directly in the integral representing the functional. We then write and solve the Euler’s equation for the new integral involving new variables. In the extremals so obtained, the new variables are changed to the original variables to get the desired extremals.

Example. Find the extremals of the functional

Sol. Let

J[r(θ)] =

z

z

θ2

θ1

θ2

θ1

r 2 + r ′ 2 dθ , where r = r(θ).

r 2 + r ′ 2 dθ .

Let

x = r cos θ, y = r sin θ.

r=

−1 x 2 + y2 , θ = tan

y x

rx + ry y′ dr dr/ dx

=

= r′ =

= dθ dθ/ dx θ x + θ y y′

x

2

x +y

1 y2

1+ 2 x

2

+

FG − y IJ

H xK

2

y

y′ x + y2

1

1

+ y′

2 x y

1+ 2 x

2

FG IJ

H K

x + yy′ x2 + y2

= xy′ − y x2 + y2

=

Also,

dθ =

xy′ − y dθ dx

. dx = (θx + θy y′)dx = 2 dx x + y2

r2 + r′2 = x2 + y2 +

= (x2 + y2)

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Medium 9781576336830

Addition and Subtraction: Decimals: Praxis I Arithmetic

Ace Academics Ace Academics ePub
Medium 9780596101640

Hack #40. Play in the Black in Blackjack

Bruce Frey O'Reilly Media ePub

Perhaps the most potentially profitable application of statistics hacking is at the blackjack table.

In blackjack, the object of the game is to get a hand of cards that is closer to totaling 21 points (without going over) than the dealer's cards. It's a simple game, really. You start with two cards and can ask for as many more as you would like. Cards are worth their face value, with the exception of face cards, which are worth 10 points, and Aces, which can be either 1 or 11.

You lose if you go over 21 or if the dealer is closer than you (without going over). The bets are even money, with the exception of getting a blackjack: two cards that add up to 21. Typically, you get paid 3-to-2 for hitting a blackjack. The dealer has an advantage in that she doesn't have to act until after you. If you bust (go over 21), she wins automatically.

Statisticians can play this game wisely by using two sources of information: the dealer's face-up card and the knowledge of cards previously dealt. Basic strategies based on probability will let smart players play almost even against the house without having to pay much attention or learn complicated systems. Methods of taking into account previously dealt cards are collectively called counting cards, and using these methods allows players to have a statistical advantage over the house.

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Medium 9781576336946

Quadrilaterals: PSAT Geometry

Ace Academics Ace Academics ePub
Medium 9781576337783

Ratio and Proportion: SAT Math II Algebra I

Ace Academics Ace Academics ePub
Medium 9781576336403

Triangles: GED Geometry

Ace Academics Ace Academics ePub
Medium 9781576337363

Sets: CLEP Algebra I

Ace Academics Ace Academics ePub
Medium 9788131807781

qtc12-1

N.P.Bali; P.N.Gupta; Dr C.P.Gandhi Laxmi Publications PDF

Unit-III

Correlation Analysis

12

Correlation and Regression

12.1. UNIVARIATE DISTRIBUTIONS

Distributions involving one variable are called univariate distributions. For example, marks obtained in a particular subject, wages earned by a group of workers.

12.2. BIVARIATE DISTRIBUTIONS

Distributions involving two variables are called bivariate distributions. For example, heights and weights of a certain group of persons, ages of husband and wife at the time of marriage.

12.3. CORRELATION

(P.T.U., M.B.A. Dec., 2009)

In a bivariate distribution, if the change in one variable affects a change in the other variable, the variables are said to be correlated.

12.4. POSITIVE OR NEGATIVE CORRELATION

(P.T.U., M.B.A. Dec. 2006)

If the increase (or decrease) in one results in a corresponding increase (or decrease) in the other, correlation is said to be direct or positive. For example, the correlation between income and expenditure is positive.

If the two variables deviate in opposite directions, i.e., if the increase (or decrease) in one results in a corresponding decrease (or increase) in the other, correlation is said to be inverse or negative. For example, the correlation between volume and pressure of a gas, correlation between price and demand, are negative correlation.

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Medium 9781576336724

Multiplication: Whole Numbers: SAT Arithmetic

Ace Academics Ace Academics ePub
Medium 9780596101640

Hack #35. Gamble Smart

Bruce Frey O'Reilly Media ePub

Why risk more than you have to when you take risks? Casino games require you to take some chances, but this chapter of real-world stat hacks will help you keep your edge and perhaps even overcome the house's edge.

Start with Texas Hold 'Em poker [Hack #36]. (Maybe you've heard of it?) When you play poker [Hack #37], play the odds [Hack #38].

Make sure, of course, to always gamble smart [Hack #35], regardless of what you play, though when it comes to the level of risk you take, some games [Hacks #39 and #40] are better than others [Hack #41].

If you like to make friendly wagers with friends or strange wagers with strangers, you can use the power of statistics to win some surprisingly winnable bar bets with cards [Hacks #42 and #44], dice [Hack #43], or just about anything else you can think of [Hack #46], including your friends' birthdays [Hack #45].

Speaking of weird gambling games (and I think we were), there are some odd statistical quirks [Hacks #47 and #49] you'll need to know when you play them, even if it is just flipping a coin [Hack #48].

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Medium 9781449370787

5. Odds and Addends

Allen B. Downey O'Reilly Media ePub

One way to represent a probability is with a number between 0 and 1, but that’s not the only way. If you have ever bet on a football game or a horse race, you have probably encountered another representation of probability, called odds.

You might have heard expressions like “the odds are three to one,” but you might not know what they mean. The odds in favor of an event are the ratio of the probability it will occur to the probability that it will not.

So if I think my team has a 75% chance of winning, I would say that the odds in their favor are three to one, because the chance of winning is three times the chance of losing.

You can write odds in decimal form, but it is most common to write them as a ratio of integers. So “three to one” is written .

When probabilities are low, it is more common to report the odds against rather than the odds in favor. For example, if I think my horse has a 10% chance of winning, I would say that the odds against are .

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Medium 9781576337387

Oblique Triangles: CLEP Precalculus

Ace Academics Ace Academics ePub

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