Let
(
𝐴
1
,
𝐴
2
,
𝐴
3
,
𝐴
4
)
be events with
𝑃
(
𝐴
𝑖
)
=
0.3
. For each (
𝑖
). Define

𝑌
=
𝐼
𝐴
1
+
𝐼
𝐴
2
+
𝐼
𝐴
3
+
𝐼
𝐴
4
.

No independence assumptions are given. Which statement must be true?

Question

Let 
(
𝐴
1
,
𝐴
2
,
𝐴
3
,
𝐴
4
)
 be events with 
𝑃
(
𝐴
𝑖
)
=
0.3
. For each (
𝑖
). Define

𝑌
=
𝐼
𝐴
1
+
𝐼
𝐴
2
+
𝐼
𝐴
3
+
𝐼
𝐴
4
.

No independence assumptions are given. Which statement must be true?

BlackTom AI · Accepted Answer

Linearity of expectation gives the quickest route: E[Y] equals the sum of the probabilities of the events, regardless of any dependence.

1) Relevant knowledge points and concepts
- Indicator random variable I_A has expectation E[I_A] = P(A).
- Linearity of expectation: E[sum X_i] = sum E[X_i] holds without any independence assumptions.
- The binomial distribution and the formula P(Y=0)=0.7^4 and Var(Y)=4·0.3·0.7 all require independence (and identical trials for binomial).

2) Step-by-step reasoning
Y = I_{A1} + I_{A2} + I_{A3} + I_{A4}. Use linearity of expectation:
E[Y] = E[I_{A1}] + E[I_{A2}] + E[I_{A3}] + E[I_{A4}].

3) Intermediate calculations
For each i, E[I_{Ai}] = P(A_i) = 0.3. Therefore
E[Y] = 0.3 + 0.3 + 0.3 + 0.3 = 4 × 0.3 = 1.2.

4) Why the correct answer is correct
Linearity of expectation does not require independence, so summing the individual expectations (each equal to 0.3) is valid and yields 1.2. This fact makes E[Y] = 1.2 necessarily true under the given information.

5) Why the other options are incorrect
- "Y ~ Bin(4,0.3)." A binomial model requires independent trials (and identical success probability). Independence is not given, so we cannot conclude a binomial distribution.
- "P(Y = 0) = 0.7^4." This equals P(no A_i) only if the complements are independent; without independence, P(A1^c ∩ ... ∩ A4^c) is not determined by the marginals alone.
- "Var(Y) = 4(0.3)(0.7)." That variance formula assumes independence (Var of sum equals sum of variances). With unknown covariances, Var(Y) could be larger or smaller; for example, if all events are identical, Var(Y) = 16·0.21 = 3.36, not 0.84.

6) Conclusion
The only statement that must be true is E[Y] = 1.2, which matches the provided correct answer.

Let ( 𝐴 1 , 𝐴 2 , 𝐴 3 , 𝐴 4 ) be events with 𝑃 ( 𝐴 𝑖 ) = 0.3 . For each ( 𝑖 ). Define 𝑌 = 𝐼 𝐴 1 + 𝐼 𝐴 2 + 𝐼 𝐴 3 + 𝐼 𝐴 4 . No independence assumptions are given. Which statement must be true?Single choice

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