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The binomial theorem describes the algebraic expansion of powers of a binomial: that is, the expansion of an expression of the form (x + y)^n where x and y are variables and n is the power to which the binomial is raised. When first encountered, n is a positive integer, but the binomial theorem can be extended to cover values of n which are fractional or negative (or both).
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Limitations of work energy theorem?
it doesn't define direction of velocity
Why Central Limit Theorem rolling a dice once you will never have a bell shaped curve?
Rolling a die once will give one point. A single point does not even define a line, let alone a curve!
Can one formally define a division ring as a field that isn't necessarily commutative?
wedderburn's little theorem says all finite division rings are commutative so they are fields. So if it is a finite division ring, then the answer is NO But for an infinite division ring... I think you can!
Can the Pythagoras theorem be proven emperically?
ANSWERYes The Pythagorean Theorem Can Be Proven Empirically.HOW?First, Lets Define The Theorem:In simplest terms, the Pythagorean Theorem is essentially a Formula that is TRUE for ANY/ALL RIGHT TRIANGLES (ANY Triangle that has ONE 90o ANGLE). The formula States: A2 + B2 = C2 , WHERE C is Always The Longest Side (Called The Hypotenuse) and is Always OPPOSITE the 90o Angle. A and B are The Other two sides of the triangle (not the Hypotenuse), the sides adjacent to the 90o Angle. To Prove The Pythagorean Theorem Empirically:First off lets define Empirically; all that it means, in this instance, is Show or Prove that the Theorem works through experience/experiment. This is very easy, just do the following: Using a protractor make a 90o AngleDraw 2 lines (Sides A & B) that make up the 90o Angle you measured out in Step 1Draw Side A - 5 cm in lengthDraw Side B - 8 cm in lengthDraw Side C - the Hypotenuse (A line that Connects Sides A and B) - But Do NOT Measure This with your ruler YET.Now since we need to PROVE that the Theorem is Correct, We have to Plug the length of the Sides A and B into the Theorem's Formula.52 + 82 = C2 (WHERE C2 is the Length of Side C/The Hypotenuse Squared)So Now we have the Equation: C2 = 25 + 64 = 89Now we need to Find what C equals, we do this by taking the Square Root of 89, and Since we know C is a Positive Number (Since its the Length of Side C), we can ignore the Negative portion of the Square Root and So We Know:C = 9.434 cmLAST STEP, NOW You MEASURE - with your Ruler, the Hypotenuse (Side C), and you will see that it equals 9.434 cm; therefore we have just Proved Empirically that the Pythagorean Theorem is Correct.
What conditions can the normal cube be used to approximate the binomial distribution?
You need to define one single variable, based on the six possible outcomes, such that the outcome of each trial is either a success or not. Thus you could define X as "roll a 5" so that the probability of success is 1/6, Or that X is "roll an even number", so the probability of success is 1/2 , or some other event. The die need not be fair, but if it is loaded, the loading must not change. This can allow you to increase the range of probabilities of "success". You then need to roll the die many times and record whether or not your chosen event occurred or not. The number of times the event occurred divided by the number of rolls will approximate a binomial distribution.
