Graduate · Rare events and convergence · 501 of 650
Rare events and convergence · Magnitude exponent a=1.75; Rarity exponent b=1.25
Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero. Givens: Magnitude exponent a=1.75; Rarity exponent b=1.25.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
For each integer n≥1, Xₙ=n^1.75 with probability n^(−1.25) and is zero otherwise. Use n=18 for the finite calculation, then consider n→∞. Find the nonzero probability, nonzero size, and mean absolute value; classify convergence of that mean to zero.
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A starting point
Multiply the nonzero magnitude and its probability. The sign of a−b decides whether n^(a−b) decreases, stays constant, or grows.
Work through the reasoning
Step 1
Identify the model and target
For each integer n≥1, Xₙ=n^1.75 with probability n^(−1.25) and is zero otherwise. Use n=18 for the finite calculation, then consider n→∞. The governing relation is Xₙ=nᵃ with probability n⁻ᵇ, else 0; E|Xₙ|=n^(a−b). Multiply the nonzero magnitude and its probability. The sign of a−b decides whether n^(a−b) decreases, stays constant, or grows.
Step 2
Substitute and calculate
At n=18, probability is (18)^(−1.25)=0.026972 and magnitude is (18)^1.75=157.299334. Their product is n^(a−b)=(18)^(1.75−1.25)=4.242641.
Step 3
Check the mathematical meaning
Here a−b=0.5, so E|Xₙ| grows without bound, so L¹ convergence fails. Since b>0, convergence in probability to zero holds. Also ∑n^(−b)<∞, so Borel–Cantelli gives almost-sure convergence for any common-space coupling. Index n: 18; Nonzero probability: 0.027; Nonzero magnitude: 157.299; Mean absolute value: 4.243. Decimal values are rounded, so use unrounded intermediate values.
The answer
At n=18, probability is (18)^(−1.25)=0.026972 and magnitude is (18)^1.75=157.299334. Their product is n^(a−b)=(18)^(1.75−1.25)=4.242641. Here a−b=0.5, so E|Xₙ| grows without bound, so L¹ convergence fails. Since b>0, convergence in probability to zero holds. Also ∑n^(−b)<∞, so Borel–Cantelli gives almost-sure convergence for any common-space coupling. Animation check: Index n: 18; Nonzero probability: 0.027; Nonzero magnitude: 157.299; Mean absolute value: 4.243. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
An event can become rare while its size increases. The nonzero probability tends to zero whenever b>0, giving convergence in probability to zero, but the mean absolute value depends on the competition between a and b. The same family can therefore converge in L¹, fail with a constant mean, or have a growing mean. For each integer n≥1, Xₙ=n^1.75 with probability n^(−1.25) and is zero otherwise. Use n=18 for the finite calculation, then consider n→∞. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
Xₙ=nᵃ with probability n⁻ᵇ, else 0; E|Xₙ|=n^(a−b)
03 · Reflect and transfer
Explain what changes and why.
How can convergence in probability coexist with a nonvanishing mean absolute error? Explain the competition between event size and rarity.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.