February 23
... (x+y+z)^n = sum (n \multichoose a,b,c) x^a y^b z^c, where the summation extends over all non-negative integers a,b,c with a+b+c=n. Section 5.6: Newton's binomial theorem Note that (n \choose 2) = n(n-1)/2, and that this makes sense even when n is not an integer. More generally, one can define (r \c ...
... (x+y+z)^n = sum (n \multichoose a,b,c) x^a y^b z^c, where the summation extends over all non-negative integers a,b,c with a+b+c=n. Section 5.6: Newton's binomial theorem Note that (n \choose 2) = n(n-1)/2, and that this makes sense even when n is not an integer. More generally, one can define (r \c ...
What are the Eigenvalues of a Sum of Non
... Notice what’s easy and what’s hard The even terms are still easy The odd terms are still easy The sum is anything but ...
... Notice what’s easy and what’s hard The even terms are still easy The odd terms are still easy The sum is anything but ...
Math 475 Big Problems, Batch 2 Big Problem 7: Tulie Number
... 1. Write down the Taylor Series for the function ex . You can look it up... I’ll assume you derived it in Calculus 2 and can prove it converges everywhere. 2. Write down an infinite series expression for e. 3. Assuming e is genuinely equal to the infinite series, prove e is irrational in the followi ...
... 1. Write down the Taylor Series for the function ex . You can look it up... I’ll assume you derived it in Calculus 2 and can prove it converges everywhere. 2. Write down an infinite series expression for e. 3. Assuming e is genuinely equal to the infinite series, prove e is irrational in the followi ...
Abstract
... A natural generalization of base B expansions is Zeckendorf’s Theorem, which states that every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers {Fn }, with Fn+1 = Fn + Fn−1 and F1 = 1, F2 = 2. If instead we allow the coefficients of the Fibonacci numbers in the decomposi ...
... A natural generalization of base B expansions is Zeckendorf’s Theorem, which states that every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers {Fn }, with Fn+1 = Fn + Fn−1 and F1 = 1, F2 = 2. If instead we allow the coefficients of the Fibonacci numbers in the decomposi ...
Discrete and Continuous Random Variables
... Suppose that the median structural capacity of a pile, under certain lateral support conditions, is 60 kN. Three piles are selected randomly for testing. Let X be the number of piles having strength under 60 kN, from amongst the three selected. The random variable X can have value 0, 1, 2, or 3. If ...
... Suppose that the median structural capacity of a pile, under certain lateral support conditions, is 60 kN. Three piles are selected randomly for testing. Let X be the number of piles having strength under 60 kN, from amongst the three selected. The random variable X can have value 0, 1, 2, or 3. If ...
Question - Advantest
... Of course, the mathematics of a Gaussian distribution imply that the tails of the curve are infinite which we know is not true because it would not be possible in the real world. However, one needs to understand that the tails of the Gaussian distribution reduce to very low probability values very ...
... Of course, the mathematics of a Gaussian distribution imply that the tails of the curve are infinite which we know is not true because it would not be possible in the real world. However, one needs to understand that the tails of the Gaussian distribution reduce to very low probability values very ...
3.6 Indicator Random Variables, and Their Means and Variances
... certain derivations. But for now, let’s establish its properties in terms of mean and variance. Handy facts: Suppose X is an indicator random variable for the event A. Let p denote P(A). Then E(X) = p ...
... certain derivations. But for now, let’s establish its properties in terms of mean and variance. Handy facts: Suppose X is an indicator random variable for the event A. Let p denote P(A). Then E(X) = p ...
Exam 1 Review 1. Describe visually, symbolically, and verbally two
... 3. From our algebra experience, we know that (x + 3) × (x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 . Explain how this “foiling” technique can be described in terms of (a) a rectangular array model, and (b) the distributive property of multiplication over addition. 4. Determine whether the set of all ...
... 3. From our algebra experience, we know that (x + 3) × (x + 2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6 . Explain how this “foiling” technique can be described in terms of (a) a rectangular array model, and (b) the distributive property of multiplication over addition. 4. Determine whether the set of all ...
Proof of the Law of Large Numbers in the Case of Finite Variance
... PROOF OF THE LAW OF LARGE NUMBERS IN THE CASE OF FINITE VARIANCE THEOREM (The Law of Large Numbers) Suppose X1 , X2 , . . . are i.i.d. random variables, each with expected value µ. Then for every > 0, ...
... PROOF OF THE LAW OF LARGE NUMBERS IN THE CASE OF FINITE VARIANCE THEOREM (The Law of Large Numbers) Suppose X1 , X2 , . . . are i.i.d. random variables, each with expected value µ. Then for every > 0, ...
PATTERNS, CONTINUED: EXPLICIT FORMULAS
... An explicit formula is a formula that enables you to compute any given term in a pattern without knowing the previous term; that is, a formula giving the nth term in terms of n. In the example of {4, 11, 18, 25, 32…}, we might use the formula 4 + 7(n -1) to find the nth term. 1. Recall the sequence ...
... An explicit formula is a formula that enables you to compute any given term in a pattern without knowing the previous term; that is, a formula giving the nth term in terms of n. In the example of {4, 11, 18, 25, 32…}, we might use the formula 4 + 7(n -1) to find the nth term. 1. Recall the sequence ...
LAB 3
... In this experiment, the final location where a ball landed is determined by the number of right and left scatterings. There are 14 rows of staggered pins. This is the n in the C.L.T. equation. Each scattering contributes a deviation, Yn, from the center horizontal location where the ball is released ...
... In this experiment, the final location where a ball landed is determined by the number of right and left scatterings. There are 14 rows of staggered pins. This is the n in the C.L.T. equation. Each scattering contributes a deviation, Yn, from the center horizontal location where the ball is released ...
Third Assignment: Solutions 1. Since P(X(p) > n) = (1 − p) n, n = 0,1
... 8. (i) If ϕ ∈ C[0, 1] then ϕ(g(ϕ)) = 0. Let τ := g(ϕ). If τ < 1 and there is ε > 0 such that ϕ > 0 on (τ − ε, τ ) and ϕ < 0 on (τ, τ + ε) then g is continuous at ϕ. To see this, let ϕn be a sequence of continuous functions such that ϕn → ϕ, uniformly. Then ϕn will eventually be positive on (τ − ε, ...
... 8. (i) If ϕ ∈ C[0, 1] then ϕ(g(ϕ)) = 0. Let τ := g(ϕ). If τ < 1 and there is ε > 0 such that ϕ > 0 on (τ − ε, τ ) and ϕ < 0 on (τ, τ + ε) then g is continuous at ϕ. To see this, let ϕn be a sequence of continuous functions such that ϕn → ϕ, uniformly. Then ϕn will eventually be positive on (τ − ε, ...
