Stats question, how to get expected values

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In summary, the conversation discusses using a random sample to estimate the mean, variance, and other parameters of a population. The equations and formulas for finding these estimators are discussed, including the concept of unbiasedness.
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wtmoore
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Homework Statement



I have taken a random sample such that X~N(8,2).

I want to use the samples that I have generated to estimate E(xbar), E(s2), E(α21) and E(α22) for the population.

Homework Equations


The Attempt at a Solution



I am not entirely sure how to do these.

I know that for a random variable, E(x)= (x1 + x2 ... +xn)/n, but I am not sure if this is the same for a sample/xbar.

I have googled and can't find the formulas, does anyone know them here?

I know for s^2 that E(s^2) = E[{sum(i to n) (xi-xbar)^2}/(n-1)]

so now I can bring the 1/(n-1) out front of the E, but I'm not sure how the summation acts.
 
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You are confusing some concepts. You want to estimate say, the mean of the random variable X μ=E{X}, you use the estimator [itex]\bar{x}=\frac{1}{N}\sum_nx_n[/itex] where x_n are the samples of X. You can prove [itex]E[\bar{x}]=\mu[/itex] so that the estimator which is a function of the samples has an expected value that is equal to the mean μ of X, so that this estimator is unbiased. Now try to find unbiased estimators of other parameters of the random variable X that you want to estimate, as a function of the samples, i.e., the expected values are equal to the parameters you want to estimate.
 

FAQ: Stats question, how to get expected values

1. What is the purpose of calculating expected values in statistics?

The purpose of calculating expected values in statistics is to determine the average outcome or value that can be expected from a particular event or data set. This helps in making predictions and understanding the probability of different outcomes.

2. How do you calculate expected values?

Expected values can be calculated by multiplying each possible outcome by its probability and then summing up all the products. This formula can be written as E(x) = Σ (x * P(x)), where x represents the possible outcomes and P(x) represents their corresponding probabilities.

3. What is the difference between expected value and actual value?

Expected value is a theoretical or predicted value while actual value is the observed or real value. Expected value is calculated based on probabilities and can be used for making predictions, while actual value is the result of an event that has already occurred.

4. Can expected values be negative?

Yes, expected values can be negative. This can happen when the possible outcomes have negative values and their corresponding probabilities are high enough to result in a negative expected value.

5. How is the concept of expected value used in decision-making?

The concept of expected value is used in decision-making to help determine the best course of action based on probabilities and potential outcomes. By calculating the expected value of each option, one can choose the option with the highest expected value to maximize their chances of a favorable outcome.

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