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A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var {Y}, where var {.} denotes the variance, equals x/64 . Find the value of x. (Answer up to the nearest integer)
Correct answer is '7'. Can you explain this answer?
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Most Upvoted Answer
A person decides to toss a fair coin repeatedly until he gets a head....
Let y = number of head
E(Y) = 7/8
= 7/8 - 49/64
= 7/64
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A person decides to toss a fair coin repeatedly until he gets a head....
Understanding the Random Variable Y
The random variable Y represents the number of heads obtained when tossing a fair coin until the first head appears, with a maximum of 3 tosses.
Possible Outcomes for Y
- If the first toss is a head (H), Y = 1.
- If the first toss is a tail (T) followed by a head (TH), Y = 1.
- If the first two tosses are tails and the third is a head (TTH), Y = 1.
- If all three tosses are tails (TTT), Y = 0.
Thus, Y can take values 0 or 1.
Probability Distribution of Y
- P(Y = 0) = P(TTT) = (1/2) * (1/2) * (1/2) = 1/8
- P(Y = 1) = P(H) + P(TH) + P(TTH) = 1/2 + (1/2) * (1/2) + (1/2) * (1/2) * (1/2) = 1/2 + 1/4 + 1/8 = 7/8
Calculating the Mean (E[Y])
E[Y] = 0 * P(Y = 0) + 1 * P(Y = 1) = 0 * (1/8) + 1 * (7/8) = 7/8
Calculating the Variance (Var[Y])
1. E[Y^2] = 0^2 * P(Y = 0) + 1^2 * P(Y = 1) = 0 + (7/8) = 7/8
2. Var[Y] = E[Y^2] - (E[Y])^2 = (7/8) - (7/8)^2 = (7/8) - (49/64) = (56/64) - (49/64) = 7/64
Conclusion
The variance of Y is given as Var[Y] = x/64. From our calculation, we found Var[Y] = 7/64, which means x = 7.
Thus, the value of x is 7.
The random variable Y represents the number of heads obtained when tossing a fair coin until the first head appears, with a maximum of 3 tosses.
Possible Outcomes for Y
- If the first toss is a head (H), Y = 1.
- If the first toss is a tail (T) followed by a head (TH), Y = 1.
- If the first two tosses are tails and the third is a head (TTH), Y = 1.
- If all three tosses are tails (TTT), Y = 0.
Thus, Y can take values 0 or 1.
Probability Distribution of Y
- P(Y = 0) = P(TTT) = (1/2) * (1/2) * (1/2) = 1/8
- P(Y = 1) = P(H) + P(TH) + P(TTH) = 1/2 + (1/2) * (1/2) + (1/2) * (1/2) * (1/2) = 1/2 + 1/4 + 1/8 = 7/8
Calculating the Mean (E[Y])
E[Y] = 0 * P(Y = 0) + 1 * P(Y = 1) = 0 * (1/8) + 1 * (7/8) = 7/8
Calculating the Variance (Var[Y])
1. E[Y^2] = 0^2 * P(Y = 0) + 1^2 * P(Y = 1) = 0 + (7/8) = 7/8
2. Var[Y] = E[Y^2] - (E[Y])^2 = (7/8) - (7/8)^2 = (7/8) - (49/64) = (56/64) - (49/64) = 7/64
Conclusion
The variance of Y is given as Var[Y] = x/64. From our calculation, we found Var[Y] = 7/64, which means x = 7.
Thus, the value of x is 7.
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A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var {Y}, where var {.} denotes the variance, equals x/64 . Find the value of x. (Answer up to the nearest integer)Correct answer is '7'. Can you explain this answer?
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