Determine the condition on coefficients "a subscript i" so that the linear combination "Summation symbol from i=1 to n for 'a subcript i'times 'X subscript i'" is an unbiased estimate of E[X].
To determine the condition on coefficients 'a subscript i' so that the linear combination is an unbiased estimate of E[X], we need to analyze the properties of unbiased estimation and the relationship between the linear combination and the expected value.
First, let's define some terms:
- 'n' represents the number of variables in the linear combination.
- 'X subscript i' represents the individual random variables that make up the linear combination.
- 'a subscript i' represents the coefficients for each random variable.
An unbiased estimator, denoted as 'Est', of a population parameter, such as the expected value E[X], satisfies the condition:
E[Est] = E[X].
Given the linear combination:
Est = Summation symbol from i=1 to n of ('a subscript i' * 'X subscript i').
To check if this linear combination is an unbiased estimate of E[X], we need to evaluate E[Est] and set it equal to E[X].
Step 1: Evaluate E[Est]
Using the linearity property of expected value, we have:
E[Est] = E[Summation symbol from i=1 to n of ('a subscript i' * 'X subscript i')]
= Summation symbol from i=1 to n of ('a subscript i' * E['X subscript i']).
Step 2: Set E[Est] equal to E[X]
To satisfy the condition E[Est] = E[X], we need to set the expression from Step 1 equal to E[X] and solve for the condition on coefficients 'a subscript i'.
Setting E[Est] equal to E[X], we get:
Summation symbol from i=1 to n of ('a subscript i' * E['X subscript i']) = E[X].
Since 'X subscript i' represents the random variables, E['X subscript i'] denotes the expected value of each random variable. So, we can rewrite the above equation as:
Summation symbol from i=1 to n of ('a subscript i' * E['X subscript i']) = E[X].
The condition on coefficients 'a subscript i' for the linear combination to be an unbiased estimator of E[X] is:
Summation symbol from i=1 to n of ('a subscript i' * E['X subscript i']) = E[X].
Note: The specific values of 'a subscript i' and 'E['X subscript i']' may vary depending on the context of the problem.