Question 1174436: You do a following experiment: You toss a biased coin, with 1/3 probability for heads and 2/3 for tails. If the coin falls heads you choose uniformly from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, if it turns out tails you choose randomly from {-1, -2, -3, -4, -5, -6}. Let X denote the random variable denoting your final outcome in this two stage experiment. Find variance of X. (Give only the variance of X as an answer, 2 decimals is enough, no other text!)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down the calculation of the variance of X step by step.
**1. Define the Random Variables and Probabilities**
* **Coin Toss:**
* Let H be the event of getting heads, with P(H) = 1/3.
* Let T be the event of getting tails, with P(T) = 2/3.
* **Choice of Numbers:**
* If heads (H), we choose a number from {1, 2, ..., 10} uniformly. Let's call this random variable Y.
* If tails (T), we choose a number from {-1, -2, ..., -6} uniformly. Let's call this random variable Z.
* X is the random variable representing the final outcome.
**2. Calculate the Expected Values**
* **E[Y] (Expected value if heads):**
* E[Y] = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) / 10 = 55 / 10 = 5.5
* **E[Z] (Expected value if tails):**
* E[Z] = (-1 - 2 - 3 - 4 - 5 - 6) / 6 = -21 / 6 = -3.5
* **E[X] (Overall expected value):**
* E[X] = E[X|H]P(H) + E[X|T]P(T) = E[Y]P(H) + E[Z]P(T)
* E[X] = (5.5)(1/3) + (-3.5)(2/3) = 5.5/3 - 7/3 = -1.5/3 = -0.5
**3. Calculate the Second Moments**
* **E[Y²] (Second moment if heads):**
* E[Y²] = (1² + 2² + ... + 10²) / 10 = (385) / 10 = 38.5
* **E[Z²] (Second moment if tails):**
* E[Z²] = ((-1)² + (-2)² + ... + (-6)²) / 6 = (91) / 6 = 15.1667 (approximately)
* **E[X²] (Overall second moment):**
* E[X²] = E[Y²]P(H) + E[Z²]P(T)
* E[X²] = (38.5)(1/3) + (91/6)(2/3) = 38.5/3 + 91/9 = 12.8333 + 10.1111 = 22.9444 (approximately)
**4. Calculate the Variance**
* **Var(X) = E[X²] - (E[X])²**
* Var(X) = 22.9444 - (-0.5)² = 22.9444 - 0.25 = 22.6944
**5. Round to Two Decimal Places**
* Var(X) ≈ 22.69
Therefore, the variance of X is approximately 22.69.
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