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If E denotes expectation, the variance of a random variable X is given by
- a)E [X2] – E2[X]
- b)E[X2] + E2[X]
- c)E [X2]
- d)E2[X]
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If E denotes expectation, the variance of a random variable X is given...
The variance is a measure of how spread out the values of X would be if the experiment leading to X were repeated a number of times.
The variance of X, written as Var(X) is given by:
Var(X) = E(X^2) − (E(X))^2
If we write E(X) = μ then:
Var(X) = E(X^2) − μ^2
Or:
Var(X) = E(X − μ)^2
This tells us that Var(X) ≥ 0
Most Upvoted Answer
If E denotes expectation, the variance of a random variable X is given...
Variance is a measure of how spread out the values of a random variable are around the expected value. It quantifies the average of the squared differences between each value and the expected value. In mathematical terms, the variance of a random variable X is denoted as Var(X) or σ².
To understand why the correct answer is option 'A', let's break down the components of the variance formula.
1. The Expected Value (E):
The expected value, denoted as E[X] or μ, represents the average value of a random variable. It is calculated by summing the product of each possible value of X and its corresponding probability.
2. The Squared Value (X²):
To find the variance, we need to calculate the squared value of each possible outcome of X. This is done by multiplying each value by itself (X * X).
3. The Expectation of the Squared Value (E[X²]):
The expectation of the squared value, denoted as E[X²], represents the average of the squared values. It is calculated in the same way as the expected value, but using the squared values instead of the original values.
Now, let's analyze the options provided:
a) E[X²] - E²[X]:
This is the correct answer. It follows the formula for calculating the variance, which subtracts the square of the expected value from the expectation of the squared value. This accounts for the spread of values around the expected value.
b) E[X²] - E²[X]:
This is not the correct answer. It only calculates the expectation of the squared value without subtracting the square of the expected value. This would not account for the spread of values and would result in a larger value than the actual variance.
c) E[X²]:
This is not the correct answer. It only calculates the expectation of the squared value without subtracting the square of the expected value. This would not account for the spread of values and would result in a larger value than the actual variance.
d) E²[X]:
This is not the correct answer. It only calculates the square of the expected value without considering the squared values of the random variable. This would not account for the spread of values and would result in a smaller value than the actual variance.
In conclusion, the correct formula for calculating the variance of a random variable X is E[X²] - E²[X], as stated in option 'A'. This formula takes into account both the spread of values around the expected value and the squared values of X.
To understand why the correct answer is option 'A', let's break down the components of the variance formula.
1. The Expected Value (E):
The expected value, denoted as E[X] or μ, represents the average value of a random variable. It is calculated by summing the product of each possible value of X and its corresponding probability.
2. The Squared Value (X²):
To find the variance, we need to calculate the squared value of each possible outcome of X. This is done by multiplying each value by itself (X * X).
3. The Expectation of the Squared Value (E[X²]):
The expectation of the squared value, denoted as E[X²], represents the average of the squared values. It is calculated in the same way as the expected value, but using the squared values instead of the original values.
Now, let's analyze the options provided:
a) E[X²] - E²[X]:
This is the correct answer. It follows the formula for calculating the variance, which subtracts the square of the expected value from the expectation of the squared value. This accounts for the spread of values around the expected value.
b) E[X²] - E²[X]:
This is not the correct answer. It only calculates the expectation of the squared value without subtracting the square of the expected value. This would not account for the spread of values and would result in a larger value than the actual variance.
c) E[X²]:
This is not the correct answer. It only calculates the expectation of the squared value without subtracting the square of the expected value. This would not account for the spread of values and would result in a larger value than the actual variance.
d) E²[X]:
This is not the correct answer. It only calculates the square of the expected value without considering the squared values of the random variable. This would not account for the spread of values and would result in a smaller value than the actual variance.
In conclusion, the correct formula for calculating the variance of a random variable X is E[X²] - E²[X], as stated in option 'A'. This formula takes into account both the spread of values around the expected value and the squared values of X.
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If E denotes expectation, the variance of a random variable X is given bya)E [X2] – E2[X] b)E[X2] + E2[X] c)E [X2] d)E2[X]Correct answer is option 'A'. Can you explain this answer?
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