Graduate · Conditional expectation by groups
Conditional expectation by groups: A small between-group gap
Conditional expectation by groups: investigate a small between-group gap with first group baseline a = 0.5; second group baseline b = 0.
Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.
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Partial information may identify a group without revealing the exact outcome. Conditional expectation must be constant on each observable group and match that group's mean. Moving from the unconditional mean toward the two group means decreases squared error, illustrating the projection interpretation in a finite space. This investigation starts with First group baseline a = 0.5; Second group baseline b = 0. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.
A relationship to keep
X=(a,a+2,b,b+4); E[X|G]=(a+1,a+1,b+2,b+2)
Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.
- STEP 1
Set up the mathematical model
Read four equally likely outcomes and the partition that reveals only whether the outcome is in the first pair or the second pair. The starting case is “A small between-group gap.”
- STEP 2
Follow the changing quantity
Move the two group estimates from the overall mean to their own group means. Original outcome values remain fixed.
- STEP 3
Explain and test the result
At the endpoint, verify that residuals sum to zero within each group and that the overall expectation is preserved. Distinguish a random conditional expectation from one scalar unconditional mean.
Your turn to explain
Make a prediction. Test your reasoning.
Keep First group baseline a = 0.5; Second group baseline b = 0. Pause the timeline at 40%. Given first group estimate = 1.65, calculate second group estimate, unconditional mean, mean squared error. Show the substitution into the displayed formula.
Use the values specified in the question. Reset restores the initial values for this investigation.
Compare your explanation
The exact group means are (0.5+0.5+2)/2=1.5 and (0+0+4)/2=2. At progress 0.4, interpolate each from overall mean 1.75, giving 1.65 and 1.85. Averaging all four squared residuals yields 2.5225. Results: Second group estimate: 1.85; Unconditional mean: 1.75; Mean squared error: 2.523. Decimal values are rounded; retain the original parameters when checking.
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