### Mean and Variance of Sample Mean | STAT /

Let X1, X2, , Xn be a random sample of size n from a distribution (population) with mean μ and variance σ2. What is the mean, that is, the expected value. The variance is computed as the average squared deviation of each number from its mean. For example, for the numbers 1, 2, and 3, the mean is 2 and the. The expectation (mean or the first moment) of a discrete random variable X is defined to be: . Variance is a measure of dispersion, telling us how “spread out” a.

You can select class width 0.

### Expectation and Variance – Mathematics A-Level Revision

The mean, variance, and standard deviation are recorded numerically in the second table. The mean and standard deviation are shown graphically as the horizontal red bar below the x-axis.

This bar is centered at the mean and extends one standard deviation on either side. In the applet, set the class width to 0.

## Moments: Mean and Variance

Compute the min, max, mean, variance, and standard deviation by hand, and verify that you get the same results as the applet. Then increase the class width to each of the other four values. A unimodal distribution that is skewed right.

A unimodal distribution that is skewed left. A symmetric bimodal distribution. Based on your experiments, answer the following questions: Is it possible for the maximum value of a distribution to be within one standard deviation of the mean? Is it possible for the minimum value of a distribution to be within one standard deviation of the mean? Is it possible for both the minimum and maximum values to be within one standard deviation of the mean? If so, try to characterize such distributions.

By experimentation, construct a distribution with the smallest possible standard deviation. Based on your result in Exercise 4, characterize the distributions with the smallest possible standard deviation. In the applet, construct a distribution that has the largest possible standard deviation.

## Mean and Variance of Sample Mean

Based on your answer to problem 6, characterize the distributions on a fixed interval [a, b] that have the largest possible standard deviation. In each case below, start with a distribution that satisfies 0. Add a point between 1 and min. Add a point between min and mean - standard deviation.

Add a point between mean - standard deviation and mean. This does not imply, however, that short term averages will reflect the mean.

Unfortunately for her, this logic has no basis in probability theory. The law of large numbers does not apply for a short string of events, and her chances of winning the next game are no better than if she had won the previous game. Properties of Means If a random variable X is adjusted by multiplying by the value b and adding the value a, then the mean is affected as follows: The new probability distribution for each outcome is provided by the following table: With the new payouts, the casino can expect to win 20 cents in the long run.

Suppose that the casino decides that the game does not have an impressive enough top prize with the lower payouts, and decides to double all of the prizes, as follows: This is equivalent to multiplying the previous value of the mean by 2, increasing the expected winnings of the casino to 40 cents.

### Variance - Wikipedia

Overall, the difference between the original value of the mean 0. The mean of the sum of two random variables X and Y is the sum of their means: Variance The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by The standard deviation is the square root of the variance. Example In the original gambling game above, the probability distribution was defined to be: Since there is not a very large range of possible values, the variance is small.

Properties of Variances If a random variable X is adjusted by multiplying by the value b and adding the value a, then the variance is affected as follows: Since the spread of the distribution is not affected by adding or subtracting a constant, the value a is not considered. And, since the variance is a sum of squared terms, any multiplier value b must also be squared when adjusting the variance.