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How to find the square of a random variable?
To find E [ θ ^ 2], you will have to find the distribution of θ ^, and then calculate its second moment. Alternatively, if you know the variance of the estimator, then E [ θ ^ 2] = V a r ( θ ^) + E [ θ ^] 2 = V a r ( θ ^) + θ 2. Thanks for contributing an answer to Cross Validated! Please be sure to answer the question.
Why do we use mean of random variable?
Mean of random variables with different probability distributions can have same values. Hence, mean fails to explain the variability of values in probability distribution. Therefore, variance of random variable is defined to measure the spread and scatter in data. Variance of a random variable is discussed in detail here on.
Is the variance of a square greater than the original variance?
The variance of the square might be greater than, less than, or equal to the original variance. (Again, it might even be infinite while the variance of the original random variable is finite.) This really should not be all that surprising. Consider a positive constant.
What happens to a random variable when the constant is equal to one?
If the constant is in (0,1), its square is less than itself. If it is equal to one, its square is equal to itself, and otherwise, its square is greater than itself. So it follows that any random variable whose distribution is entirely in [math] (0,1) [/math] will behave like the constant in that interval.
How to estimate the expected value and variance of a random variable?
I have asked this in a general way here: Approximating the expected value and variance of the function of a (continuous univariate) random variable . I have also read answers and coments to this question: Variance of powers of a random variable , but I think it refers to integer powers, which is not my case.
Which is the expected sum of X and Y?
E(X + Y) = E(X) + E(Y). (The expectation of a sum = the sum of the expectations. This rule extends as you would expect it to when there are more than 2 random variables, e.g. 9. If X and Y are independent, E(XY) = E(X)E(Y).
When to use the square root of G?
That is to say, I am interested in calculating or estimating E [ g ( X)] and V a r [ g ( X)] when g is the square root function. I have asked this in a general way here: Approximating the expected value and variance of the function of a (continuous univariate) random variable .