Assignment- 01 Course Title: Applied Categorical Data Analysis Course Code: AST-509 Submitted to: M.H.M Imrul Kabir Senior Lecturer Department of Mathematical and Physical Sciences Submitted by: MD. Shofiul Bashar Rafi ID-2019-3-82-002 Section# 01 Date: 05-07-2020 MODE for Binomial Distribution: Let, m= Mode, and p is probability of success and q is the probability of failure , also note that, p+q=1 According to concept of mood for binomial distribution, we can write: 𝑷𝒓𝒐𝒃(𝒙 = 𝒎 + 𝟏) ≤ 𝑷𝒓𝒐𝒃(𝒙 = 𝒎) ≥ 𝑷𝒓𝒐𝒃(𝒙 = 𝒎 − 𝟏) Now, 𝑃𝑟𝑜𝑏(𝑥 = 𝑚 + 1) ≤ 𝑃𝑟𝑜𝑏(𝑥 = 𝑚) ⟹ nCm+1 𝑝𝑚+1 𝑞 𝑛−𝑚−1 ≤ nCm 𝑝𝑚 𝑞 𝑛−𝑚 𝑛! ⟹ (𝑚+1)!(𝑛−𝑚−1)! 𝑝𝑚 𝑝 𝑞 𝑛−𝑚 𝑞 −1 ≤ 𝑛! 𝑚!(𝑛−𝑚)! (𝑛−𝑚)! 𝑞 𝑚! ⟹ (𝑚+1)! (𝑛−𝑚−1)! ≤ 1 𝑝 ⟹ (𝑛−𝑚) 𝑝 (𝑚+1) 𝑞 ≤1 ⟹ 𝑝𝑛 − 𝑝𝑚 ≤ 𝑞𝑚 + 𝑞 ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑞𝑚 + 𝑝𝑚 ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑚(𝑝 + 𝑞) ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑚(𝑝 + 𝑞) ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑚 × 1[ 𝑠𝑖𝑛𝑐𝑒 𝑝 + 𝑞 = 1] ⟹ 𝒎 ≥ 𝒑𝒏 − 𝒒 Again, 𝑃𝑟𝑜𝑏(𝑥 = 𝑚) ≥ 𝑃𝑟𝑜𝑏(𝑥 = 𝑚 − 1) ⟹ nCm 𝑝𝑚 𝑞 𝑛−𝑚 ≥ nCm-1 𝑝𝑚−1 𝑞 𝑛−𝑚+1 ⟹ ⟹ ⟹ 𝑛! 𝑛! 𝑚!(𝑛−𝑚)! 𝑝𝑚 𝑞 𝑛−𝑚 ≥ (𝑚−1)!(𝑛−𝑚+1)! 𝑝𝑚 𝑝 −1 𝑞 𝑛−𝑚 𝑞 (𝑚−1)!(𝑛−𝑚+1)! 𝑝 𝑚!(𝑛−𝑚)! (𝑛−𝑚+1) 𝑝 𝑚 𝑞 𝑞 ≥1 ≥1 ⟹ 𝑝𝑛 − 𝑝𝑚 + 𝑝 ≥ 𝑞𝑚 ⟹ 𝑝(𝑛 + 1) + 𝑝 ≥ 𝑚(𝑝 + 𝑞) ⟹ 𝒎 ≤ 𝒑(𝒏 + 𝟏) So, 𝑃𝑟𝑜𝑏(𝑥 = 𝑚 + 1) ≤ 𝑃𝑟𝑜𝑏(𝑥 = 𝑚) ≥ 𝑃𝑟𝑜𝑏(𝑥 = 𝑚 − 1) ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑚 ≤ 𝑝(𝑛 + 1) ⟹ 𝑝𝑛 − 𝑞 ≤ 𝑚 ≤ 𝑝(𝑛 + 1) ⟹ 𝑝(𝑛 + 1)-1 ≤ 𝑚 ≤ 𝑝(𝑛 + 1) , This is the formula of Mode of binomial distribution 𝑝𝑚 𝑞 𝑛−𝑚 Median of Binomial Distribution: There is no exact formula of Median for Binomial distribution. In many journals, researchers describe their own developed formula based on following characteristics: If, p, and n is are integer, then Median≥ mean of binomial(np) When, p=.5, and n is an odd number then median lies within the interval, 0.5(n-1)≤0.5(n+1) And when n is even number, then 𝑛 Median= 2 𝑛 ∑𝑚 0 𝑝𝑟𝑜𝑏(𝑥 = 𝑚𝑒𝑑𝑖𝑎𝑛) ≥ 0.5 𝑜𝑟 ∑𝑚 𝑝𝑟𝑜𝑏(𝑥 = 𝑚𝑒𝑑𝑖𝑎𝑛) ≥ 0.5