ꯚꯦꯌꯔꯤꯌꯦꯟꯁ

ꯕꯦꯠ ꯑꯃꯒꯤ ꯐꯣꯔꯃꯨꯂꯥ

ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯦꯊꯃꯦꯇꯤꯛꯁꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯍꯩ ꯑꯃꯥ ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ, 0 ꯗꯒꯤ 1 ꯐꯥꯑꯣꯕ ꯅꯠꯔꯒ 0% ꯗꯒꯤ 100% ꯐꯥꯑꯣꯕ꯫ ꯃꯁꯤꯡ ꯑꯃꯥ: 1/37 = 2.70%꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → p ꯒꯥ ꯂꯣꯌꯅꯅ W ꯗꯒꯤ 1 ꯄꯤꯕ ꯕꯦꯠ ꯑꯃꯗ, ꯌꯨꯅꯤꯠꯁꯦꯜꯌꯨꯅꯤꯠꯁꯤꯁ꯭ꯇꯦꯝ ꯅꯠꯔꯒ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯁꯤꯖꯤꯟꯅꯕ ꯃꯔꯨꯑꯣꯏꯕ ꯊꯝꯂꯕ ꯁꯦꯜ꯫ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡ ꯌꯨꯅꯤꯠꯇ ꯁꯟꯗꯣꯛꯂꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯃꯈꯣꯌ ꯀꯔꯤꯒꯨꯝꯕ ꯇꯦꯕꯜꯗ ꯆꯠꯂꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯈꯨꯗꯤꯡꯒꯤ ꯃꯍꯩ ꯑꯁꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ p ꯗ +W ꯑꯃꯁꯨꯡ ꯈꯨꯗꯣꯡꯆꯥꯕ 1 − p ꯗ −1 ꯅꯤ꯫ ꯐꯪꯒꯗꯕ ꯃꯃꯜꯃꯦꯊꯃꯦꯇꯤꯛꯁꯐꯪꯒꯗꯕ ꯃꯃꯜꯃꯇꯝ ꯁꯥꯡꯅ ꯌꯦꯡꯂꯕꯗ ꯕꯦꯠ ꯑꯃꯒꯤ ꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ꯫ ꯃꯔꯨꯑꯣꯏꯕ ꯌꯨꯔꯣꯄꯤꯌꯟ ꯕꯦꯠ ꯈꯨꯗꯤꯡꯗ ꯊꯝꯂꯕ ꯁꯦꯜꯒꯤ −1/37꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯁꯤ pW − (1 − p) = −1/37 ꯅꯤ꯫ ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯃꯦꯊꯃꯦꯇꯤꯛꯁꯚꯦꯌꯔꯤꯌꯦꯟꯁꯃꯍꯩꯁꯤꯡ ꯐꯪꯒꯗꯕ ꯃꯃꯂꯒꯤ ꯑꯀꯣꯌꯕꯗ ꯀꯌꯥ ꯁꯟꯗꯣꯛꯅ ꯂꯩꯕꯒꯦ ꯍꯥꯌꯕꯒꯤ ꯆꯥꯡ꯫ ꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯇ ꯋꯥꯡꯏ, ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠꯁꯤꯡꯗ ꯅꯦꯝꯏ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯁꯤ E[X²] − EV² = pW² + (1 − p) − (1/37)² ꯅꯤ꯫ ꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯕꯦꯠꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯃꯁꯤꯡ ꯑꯃꯈꯛꯇ, 35:1 ꯄꯦꯑꯥꯎꯠ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯗ: (1/37)(1225) + 36/37 − 0.0007 ≈ 34.08꯫ ꯑꯉꯥꯡꯕꯕꯦꯠꯑꯉꯥꯡꯕꯑꯉꯥꯡꯕ ꯃꯁꯤꯡ 18 ꯗ ꯊꯝꯕ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ꯫ ꯐ꯭ꯔꯦꯟꯆ: ꯔꯨꯖ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯗ: 18/37 + 19/37 − 0.0007 ≈ 0.999꯫

ꯁꯦꯁꯟ ꯑꯃꯒꯤꯗꯃꯛ ꯃꯁꯤꯅ ꯀꯔꯤ ꯍꯥꯌꯕꯒꯦ

ꯃꯔꯤ ꯂꯩꯅꯗꯕ ꯁ꯭ꯄꯤꯟꯁꯤꯡꯃꯦꯊꯃꯦꯇꯤꯛꯁꯃꯔꯤ ꯂꯩꯅꯗꯕ ꯁ꯭ꯄꯤꯟꯁꯤꯡꯍꯥꯟꯅꯒꯤ ꯃꯍꯩ ꯀꯔꯤꯒꯨꯝꯕ ꯑꯣꯏꯔꯕꯁꯨ ꯁ꯭ꯄꯤꯟ ꯈꯨꯗꯤꯡꯃꯛꯀꯤ ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ ꯃꯥꯟꯅꯩ꯫ ꯋꯦꯕꯁꯥꯏꯠ ꯑꯁꯤꯒꯤ ꯊꯤꯕ ꯈꯨꯗꯤꯡꯃꯛꯀꯤ ꯌꯨꯝꯐꯝꯅꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯗ ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯁꯤꯡ ꯄꯨꯟꯁꯤꯟꯅꯩ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯊꯝꯂꯕ ꯁꯦꯜ ꯍꯣꯡꯗꯅ ꯁ꯭ꯄꯤꯟ N ꯗ ꯄꯨꯝꯅꯃꯛꯀꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ N × ꯊꯝꯂꯕ ꯁꯦꯜ² × σ² ꯑꯣꯏ, ꯑꯃꯁꯨꯡ ꯁ꯭ꯇꯦꯟꯗꯔꯗ ꯗꯤꯚꯤꯌꯦꯁꯟꯃꯦꯊꯃꯦꯇꯤꯛꯁꯁ꯭ꯇꯦꯟꯗꯔꯗ ꯗꯤꯚꯤꯌꯦꯁꯟꯚꯦꯌꯔꯤꯌꯦꯟꯁꯀꯤ ꯁ꯭ꯀ꯭ꯋꯦꯌꯔ ꯔꯨꯠ; ꯃꯍꯩ ꯑꯃꯅ ꯐꯪꯒꯗꯕ ꯃꯃꯜꯗꯒꯤ ꯌꯥꯝꯅ ꯂꯥꯞꯄ ꯃꯑꯣꯡ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → σ × ꯊꯝꯂꯕ ꯁꯦꯜ × √N ꯑꯣꯏ꯫ $10 ꯒꯤ ꯁ꯭ꯄꯤꯟ 100: ꯁ꯭ꯇꯦꯟꯗꯔꯗ ꯗꯤꯚꯤꯌꯦꯁꯟ ꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯁꯤꯡꯒꯤꯗꯃꯛ $584 ꯃꯔꯛ ꯑꯃꯁꯨꯡ ꯑꯉꯥꯡꯕꯒꯤꯗꯃꯛ $100 ꯅꯤ, ꯃꯥꯟꯅꯕ ꯐꯪꯒꯗꯕ −$27 ꯃꯔꯛꯀꯤ ꯃꯅꯥꯛꯇ꯫ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯃꯍꯩꯁꯤꯡ ꯂꯤꯁꯤꯡ ꯀꯌꯥꯅ ꯆꯣꯡꯊꯕ ꯎꯏ; ꯑꯇꯣꯞꯄ ꯑꯗꯨꯅ ꯇꯞꯅ ꯆꯦꯟꯊꯕ ꯎꯏ꯫ ꯑꯅꯤꯃꯛꯅ ꯃꯃꯜ ꯑꯗꯨꯃꯛ ꯄꯤꯔꯤ꯫

ꯃꯔꯤ ꯂꯩꯅꯗꯕ ꯔꯨꯂꯦꯠ ꯁ꯭ꯄꯤꯟꯁꯤꯡ

ꯚꯦꯌꯔꯤꯌꯦꯟꯁ ꯑꯃꯁꯨꯡ ꯁꯤꯁ꯭ꯇꯦꯝꯁꯤꯡ

ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟꯊꯧꯔꯥꯡꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟꯃꯃꯥꯡꯒꯤ ꯃꯍꯩꯁꯤꯡꯒꯤ ꯃꯈꯥꯗ ꯊꯝꯂꯕ ꯁꯦꯜ ꯍꯣꯡꯍꯟꯕ ꯅꯤꯌꯝ ꯀꯔꯤꯒꯨꯝꯕ꯫ ꯅꯦꯒꯦꯇꯤꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡꯅ ꯃꯥꯏꯊꯤꯔꯕꯥ ꯃꯇꯨꯡꯗ ꯍꯦꯅꯒꯠꯍꯜꯂꯤ, ꯄꯣꯖꯤꯇꯤꯕꯁꯤꯡꯅ ꯃꯥꯏꯄꯥꯛꯂꯕꯥ ꯃꯇꯨꯡꯗ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯅ ꯊꯝꯂꯕ ꯌꯨꯅꯤꯠ ꯈꯨꯗꯤꯡꯒꯤ EV ꯍꯣꯡꯍꯟꯗꯦ ꯑꯗꯨꯕꯨ ꯊꯝꯂꯕ ꯁꯦꯜ ꯑꯃꯁꯨꯡ ꯗꯤꯁ꯭ꯇ꯭ꯔꯤꯕ꯭ꯌꯨꯁꯅꯒꯤ ꯃꯑꯣꯡ ꯑꯅꯤꯃꯛ ꯍꯣꯡꯍꯜꯂꯤ꯫ ꯃꯥꯔꯇꯤꯟꯒꯦꯜꯊꯧꯔꯥꯡꯃꯥꯔꯇꯤꯟꯒꯦꯜꯃꯥꯏꯊꯤꯕ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯇꯨꯡꯗ ꯕꯦꯠ ꯑꯗꯨ ꯑꯅꯤ ꯊꯥꯛ ꯍꯦꯅꯒꯠꯍꯟꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ꯫ ꯃꯁꯤꯅ ꯀꯔꯤ ꯇꯧꯕꯒꯦ ꯑꯃꯁꯨꯡ ꯀꯔꯤ ꯇꯧꯗꯕꯒꯦ ꯍꯥꯌꯕ ꯊꯧꯔꯥꯡꯒꯤ ꯁꯦꯛꯁꯅꯗ ꯌꯦꯡꯕꯤꯌꯨ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯅ ꯁꯦꯁꯟ ꯌꯥꯝꯅꯥ ꯀꯌꯥꯒꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ ꯍꯟꯊꯍꯜꯂꯤ (ꯑꯄꯤꯀꯄ ꯃꯥꯏꯄꯥꯛꯄ ꯀꯌꯥ) ꯑꯃꯁꯨꯡ ꯑꯆꯧꯕ ꯃꯇꯥꯏ ꯑꯃꯥ ꯍꯥꯄꯆꯤꯜꯂꯤ (ꯈꯔꯥ ꯈꯛꯇꯗ ꯊꯣꯀꯄ ꯑꯆꯧꯕ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ); ꯃꯇꯥꯏ ꯑꯗꯨꯒꯤ ꯃꯔꯃꯗꯒꯤ ꯁꯦꯁꯟ ꯄꯨꯝꯅꯃꯛꯀꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ ꯐ꯭ꯂꯦꯠ ꯕꯦꯇꯤꯡꯊꯧꯔꯥꯡꯐ꯭ꯂꯦꯠ ꯕꯦꯇꯤꯡꯁ꯭ꯄꯤꯟ ꯈꯨꯗꯤꯡꯗ ꯁꯦꯜ ꯃꯥꯟꯅꯕ ꯊꯝꯕ꯫ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ ꯈꯨꯗꯤꯡꯃꯛ ꯆꯥꯡꯗꯝꯅꯕ ꯃꯔꯨꯑꯣꯏꯕ ꯃꯈꯥꯌꯣꯜꯅꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯒꯤꯗꯒꯤ ꯍꯦꯟꯅꯥ ꯋꯥꯡꯏ꯫ ꯄꯥꯔꯣꯂꯤꯊꯧꯔꯥꯡꯄꯥꯔꯣꯂꯤꯄꯣꯖꯤꯇꯤꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ: ꯃꯥꯏꯄꯥꯛꯂꯕꯥ ꯃꯇꯨꯡꯗ ꯑꯅꯤ ꯊꯥꯛ ꯍꯦꯅꯒꯠꯍꯜꯂꯤ, ꯃꯥꯏꯊꯤꯔꯕꯥ ꯃꯇꯨꯡꯗ ꯅꯠꯔꯒ ꯃꯈꯥ ꯆꯠꯅꯅ ꯑꯃꯨꯛ ꯑꯍꯨꯝ ꯃꯥꯏꯄꯥꯛꯂꯕꯥ ꯃꯇꯨꯡꯗ ꯑꯍꯧꯕꯗ ꯍꯜꯂꯀꯏ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯔꯤꯚꯔꯁ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯅ ꯃꯁꯤꯒꯤ ꯑꯣꯟꯊꯣꯛꯄ ꯇꯧꯏ꯫ ꯁꯤꯁ꯭ꯇꯦꯝ ꯑꯃꯥ ꯈꯅꯕ ꯍꯥꯌꯕꯁꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯀꯤ ꯃꯑꯣꯡ ꯑꯃꯥ ꯈꯅꯕꯅꯤ; ꯑꯦꯚꯔꯦꯖ ꯑꯗꯨ ꯈꯅꯕ ꯌꯥꯗꯦ꯫

ꯐ꯭ꯂꯦꯠ ꯕꯦꯇꯤꯡ ꯔꯨꯂꯦꯠ ꯊꯧꯔꯥꯡꯄꯥꯔꯣꯂꯤ ꯔꯨꯂꯦꯠ ꯊꯧꯔꯥꯡ (ꯔꯤꯚꯔꯁ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ)

ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯒꯤꯗꯃꯛ ꯃꯁꯤ ꯀꯔꯤꯒꯤ ꯃꯔꯨ ꯑꯣꯏꯕꯒꯦ

ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯅ ꯃꯇꯝ ꯀꯨꯞꯄ ꯁꯦꯁꯟꯁꯤꯡ ꯈꯉꯕ ꯉꯃꯗꯕ ꯑꯃꯁꯨꯡ ꯃꯇꯝ ꯁꯥꯡꯕ ꯁꯦꯁꯟꯁꯤꯡ ꯈꯉꯕ ꯉꯝꯕ ꯑꯣꯏꯍꯜꯂꯤ꯫ ꯁ꯭ꯄꯤꯟ 100 ꯗ σ = $100 ꯑꯃꯁꯨꯡ EV = −$27 ꯑꯣꯏꯔꯕꯗ ꯑꯦꯖ ꯑꯁꯤ ꯅꯣꯏꯖꯀꯤ ꯁꯔꯨꯛ ꯃꯔꯤꯒꯤ ꯑꯃꯈꯛꯇꯅꯤ ꯑꯃꯁꯨꯡ ꯃꯥꯏꯄꯥꯛꯄ ꯁꯦꯁꯟ ꯑꯃꯥ ꯃꯍꯧꯁꯥꯒꯤꯅꯤ꯫ ꯁ꯭ꯄꯤꯟ 10,000 ꯗ EV −$2,700 ꯑꯃꯁꯨꯡ σ $1,000 ꯑꯣꯏ; ꯍꯧꯖꯤꯛ ꯑꯦꯖ ꯑꯁꯤ ꯅꯣꯏꯖꯗꯒꯤ ꯑꯃꯨꯛ ꯑꯍꯨꯝ ꯃꯔꯛ ꯍꯦꯟꯏ ꯑꯃꯁꯨꯡ ꯃꯥꯏꯄꯥꯛꯄ ꯁꯦꯁꯟ ꯑꯃꯥ ꯊ꯭ꯔꯤ-ꯁꯤꯒꯃꯥꯒꯤ ꯊꯧꯗꯣꯛ ꯑꯃꯅꯤ꯫ ꯃꯇꯝ ꯀꯨꯞꯄꯗ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ ꯑꯁꯤ ꯁꯥꯟꯅꯔꯤꯕꯒꯤ ꯃꯔꯨꯞꯅꯤ ꯑꯃꯁꯨꯡ ꯃꯇꯝ ꯁꯥꯡꯅꯗꯤ ꯀꯦꯁꯤꯅꯣꯒꯤ ꯃꯔꯨꯞꯅꯤ꯫

ꯄꯨꯟꯁꯤꯟꯕ ꯕꯦꯠꯁꯤꯡꯒꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ

ꯁ꯭ꯄꯤꯟ ꯑꯃꯈꯛꯇ ꯊꯝꯕ ꯕꯦꯠ ꯀꯌꯥꯒꯤ ꯃꯍꯩꯁꯤꯡ ꯑꯃꯒꯥ ꯑꯃꯒꯥ ꯃꯔꯤ ꯂꯩꯅꯩ (17 ꯊꯣꯀꯄꯥ ꯑꯁꯤ ꯑꯃꯨꯕꯕꯦꯠꯑꯃꯨꯕꯑꯃꯨꯕ ꯃꯁꯤꯡ 18 ꯗ ꯊꯝꯕ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ꯫ ꯐ꯭ꯔꯦꯟꯆ: ꯅꯣꯌꯥꯔ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯁꯨꯡ ꯑꯅꯤꯁꯨꯕ ꯗꯖꯟꯕꯦꯠꯗꯖꯟ1–12, 13–24 ꯅꯠꯔꯒ 25–36, 2:1 ꯄꯦꯑꯥꯎꯠ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ →ꯗꯁꯨ ꯃꯥꯏꯄꯥꯛꯄꯅꯤ), ꯃꯔꯝ ꯑꯗꯨꯅ ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯁꯤꯡ ꯌꯥꯝꯅꯥ ꯂꯥꯏꯅ ꯄꯨꯟꯁꯤꯟꯗꯦ꯫ “ꯕꯦꯠ ꯁꯦꯝꯕꯒꯤ ꯈꯨꯠꯂꯥꯏ” ꯅ ꯃꯍꯩ 37 ꯄꯨꯝꯅꯃꯛ ꯃꯁꯤꯡ ꯊꯤꯗꯨꯅ ꯆꯤꯞꯁꯤꯡꯁꯦꯜꯆꯤꯞꯂꯦꯞꯂꯕ ꯁꯦꯜꯒꯤ ꯃꯃꯜ ꯇꯥꯛꯄ ꯇꯣꯀꯟ, ꯕꯦꯠ ꯊꯝꯅꯕ ꯂꯦꯑꯥꯎꯠꯇ ꯊꯝꯃꯤ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯆꯦꯛ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯀꯔꯤꯒꯨꯝꯕ ꯃꯄꯨꯟ ꯑꯃꯒꯤ ꯑꯆꯨꯝꯕ ꯗꯤꯁ꯭ꯇ꯭ꯔꯤꯕ꯭ꯌꯨꯁꯟ ꯊꯤꯖꯤꯜꯂꯤ, ꯃꯁꯤ ꯆꯨꯃꯅ ꯇꯧꯕꯒꯤ ꯂꯝꯕꯤ ꯑꯃꯈꯛꯇꯅꯤ꯫

ꯕꯦꯠ ꯁꯦꯝꯕꯒꯤ ꯈꯨꯠꯂꯥꯏ

ꯐꯣꯔꯃꯨꯂꯥ

ꯋꯥꯍꯪꯁꯤꯡ