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The two sides of a fair coin are labelled as 0 and 1. The coin is tossed two times independently. Let M and N denote the labels corresponding to the outcomes of those tosses. For a random variable X, defined as X = min{M, N), the expected value E(X) (rounded off to two decimal places) is ________ .
Correct answer is '0.25'. Can you explain this answer?
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Verified Answer
The two sides of a fair coin are labelled as 0 and 1. The coin is toss...
s = {( H, H), (H, T), (T, H), (T, T)}
Now, X = Min {M, N}
∴ X = Min {H, H} = Min{(1, 1)} = 1
X = Min {H, T} = Min{1, 0} = 0
X = Min{T, H} = Min{0, 1} = 0
X = Min{T, T} = Min{0, 0} = 0
∴
We know that, E(X) =
Now, X = Min {M, N}
∴ X = Min {H, H} = Min{(1, 1)} = 1
X = Min {H, T} = Min{1, 0} = 0
X = Min{T, H} = Min{0, 1} = 0
X = Min{T, T} = Min{0, 0} = 0
∴
We know that, E(X) =
Most Upvoted Answer
The two sides of a fair coin are labelled as 0 and 1. The coin is toss...
Understanding the Problem:
The problem describes a fair coin that is tossed two times independently. Each toss can result in either a 0 or a 1. The labels corresponding to the outcomes of the two tosses are denoted as M and N. The random variable X is defined as the minimum of M and N.
Calculating the Expected Value:
To find the expected value E(X), we need to calculate the average value of X over all possible outcomes of the two coin tosses. We can do this by considering all possible outcomes and their probabilities.
Outcomes:
Since each coin toss can result in either a 0 or a 1, there are four possible outcomes for the two coin tosses:
- Outcome 1: M = 0, N = 0
- Outcome 2: M = 0, N = 1
- Outcome 3: M = 1, N = 0
- Outcome 4: M = 1, N = 1
Calculating the Probability:
Since the coin is fair, each outcome has an equal probability of 1/4. Therefore, the probability of each outcome is as follows:
- P(Outcome 1) = 1/4
- P(Outcome 2) = 1/4
- P(Outcome 3) = 1/4
- P(Outcome 4) = 1/4
Calculating the Minimum:
Now, let's calculate the minimum for each outcome:
- Minimum of Outcome 1: min(0, 0) = 0
- Minimum of Outcome 2: min(0, 1) = 0
- Minimum of Outcome 3: min(1, 0) = 0
- Minimum of Outcome 4: min(1, 1) = 1
Calculating the Expected Value:
To find the expected value E(X), we need to calculate the weighted average of the minimum values over all possible outcomes:
E(X) = P(Outcome 1) * Minimum of Outcome 1 + P(Outcome 2) * Minimum of Outcome 2 + P(Outcome 3) * Minimum of Outcome 3 + P(Outcome 4) * Minimum of Outcome 4
Substituting the values:
E(X) = (1/4) * 0 + (1/4) * 0 + (1/4) * 0 + (1/4) * 1
E(X) = 0 + 0 + 0 + 1/4
E(X) = 1/4
Therefore, the expected value E(X) is 0.25 when rounded off to two decimal places.
The problem describes a fair coin that is tossed two times independently. Each toss can result in either a 0 or a 1. The labels corresponding to the outcomes of the two tosses are denoted as M and N. The random variable X is defined as the minimum of M and N.
Calculating the Expected Value:
To find the expected value E(X), we need to calculate the average value of X over all possible outcomes of the two coin tosses. We can do this by considering all possible outcomes and their probabilities.
Outcomes:
Since each coin toss can result in either a 0 or a 1, there are four possible outcomes for the two coin tosses:
- Outcome 1: M = 0, N = 0
- Outcome 2: M = 0, N = 1
- Outcome 3: M = 1, N = 0
- Outcome 4: M = 1, N = 1
Calculating the Probability:
Since the coin is fair, each outcome has an equal probability of 1/4. Therefore, the probability of each outcome is as follows:
- P(Outcome 1) = 1/4
- P(Outcome 2) = 1/4
- P(Outcome 3) = 1/4
- P(Outcome 4) = 1/4
Calculating the Minimum:
Now, let's calculate the minimum for each outcome:
- Minimum of Outcome 1: min(0, 0) = 0
- Minimum of Outcome 2: min(0, 1) = 0
- Minimum of Outcome 3: min(1, 0) = 0
- Minimum of Outcome 4: min(1, 1) = 1
Calculating the Expected Value:
To find the expected value E(X), we need to calculate the weighted average of the minimum values over all possible outcomes:
E(X) = P(Outcome 1) * Minimum of Outcome 1 + P(Outcome 2) * Minimum of Outcome 2 + P(Outcome 3) * Minimum of Outcome 3 + P(Outcome 4) * Minimum of Outcome 4
Substituting the values:
E(X) = (1/4) * 0 + (1/4) * 0 + (1/4) * 0 + (1/4) * 1
E(X) = 0 + 0 + 0 + 1/4
E(X) = 1/4
Therefore, the expected value E(X) is 0.25 when rounded off to two decimal places.
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The two sides of a fair coin are labelled as 0 and 1. The coin is tossed two times independently. Let M and N denote the labels corresponding to the outcomes of those tosses. For a random variable X, defined as X = min{M, N), the expected value E(X) (rounded off to two decimal places) is ________ .Correct answer is '0.25'. Can you explain this answer?
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