First, find the corresponding row from Pascal's Triangle. Second, write decreasing powers of the first term. Third, write increasing powers of the second term. Drawing the skeleton in first and using it to organize your work helps limit mistakes. Read Binomial Theorem Practice Problems Lesson ...
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(nk)(nk) is also called the binomial coefficient. This is because the coefficients in the binomial theorem are given by (nk)(nk). In particular, the binomial theorem states that for an integer n≥0n≥0, we have (a+b)n=∑k=0n(nk)akbn−k.(a+b)n=∑k=0n(nk)akbn−k.Note...
Theorem 2.2 Let X ∼ NB-GE(r, α, β). The probability mass function of X is given by: ⎛⎞ f (x; r, α, β) = r+x−1 x x j=0 x j Γ(α + 1)Γ (−1)j ⎝ 1 + r+j βΓα + r+j β + 1 ⎠, (5) where x = 0, 1, 2, . . . , r, α ...
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Theorem 2. Suppose that X has the𝐵𝐵𝐸2(𝜆,𝜃,𝑎,𝑏), then the moment-generating function of RV X is as below 𝑀𝑋(𝑡)=∑𝑖,𝑗=0∞𝑑𝑖,𝑗Γ(𝑗+1)[𝜆(𝑏+𝑖)−𝑡]𝑗+2[2(1−𝜃)[𝜆(𝑏+𝑖)−𝑡]+𝜆𝜃(𝑗+1)]. (13) Pr...
generally meaning finding a probability below 0.05 for their particular hypothesis test. Not only is this a dangerous practice in which results which are important but show no statistically significant difference in, for example, treatment versus control are not even published, but also many times th...
Theorem 4. Let G ∈ C tan σ , m , n , δ , υ . Then, G is convex of order β , 0 ≤ β < 1 and ζ < r 1 , where r 1 = inf j ≥ 2 1 − β 4 j − 2 2 + tan e i θ ψ j j − β tan e i θ 1 j − 1 . (26) Proof. It is sufficient...
Theorem 2. Suppose that X has the𝐵𝐵𝐸2(𝜆,𝜃,𝑎,𝑏), then the moment-generating function of RV X is as below 𝑀𝑋(𝑡)=∑𝑖,𝑗=0∞𝑑𝑖,𝑗Γ(𝑗+1)[𝜆(𝑏+𝑖)−𝑡]𝑗+2[2(1−𝜃)[𝜆(𝑏+𝑖)−𝑡]+𝜆𝜃(𝑗+1)]. (13) Pr...