expected value

(noun)

of a discrete random variable, the sum of the probability of each possible outcome of the experiment multiplied by the value itself

Related Terms

Examples of expected value in the following topics:

  • Expected Values of Discrete Random Variables

    • The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
    • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
    • The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
    • This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average.
    • The intuition, however, remains the same: the expected value of $X$ is what one expects to happen on average.

  • Expected Value and Standard Error

    • In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
    • The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity.
    • The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
    • If we were to selected one number from the box, the expected value would be:
    • The new expected value of the sum of the numbers can be calculated by the number of draws multiplied by the expected value of the box: $25\cdot 2.2 = 55$.

  • Expected Value

    • The expected value is a weighted average of all possible values in a data set.
    • In probability theory, the expected value refers, intuitively, to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained.
    • More formally, the expected value is a weighted average of all possible values.
    • The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
    • The expected value plays important roles in a variety of contexts.

  • The Correction Factor

    • The expected value is a weighted average of all possible values in a data set.
    • In probability theory, the expected value refers, intuitively, to the value of a random variable one would "expect" to find if one could repeat the random variable process an infinite number of times and take the average of the values obtained.
    • More formally, the expected value is a weighted average of all possible values.
    • The value may not be expected in the ordinary sense—the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
    • The expected value plays important roles in a variety of contexts.

  • Mean or Expected Value and Standard Deviation

    • The expected value is often referred to as the "long-term"average or mean .
    • The expected value is 1.1.
    • The expected value is the expected number of times a newborn wakes its mother after midnight.
    • Add the last column to find the expected value. µ = Expected Value = 105/50 = 2.1
    • What is the expected value, µ?

  • Summary of Formulas

    • The test statistic is $\sum_k \frac{(OE)^2}{ E}$ , where O = observed values (data), E = expected values (from theory), and k = the number of different data cells or categories.
    • The test statistic is $\displaystyle \sum_{i \cdot j} \frac{(OE)^2}{ E}$ where O=observed values, E=expected values, i =the number of rows in the table, and j = the number of columns in the table.
    • The test statistic is $\sum_{i \cdot j} \frac{(OE)^2}{ E}$ where O=observed values, E=expected values, i =the number of rows in the table, and j = the number of columns in the table.
    • If the null hypothesis is true, the expected number E = (row total)(column total) total surveyed .
    • NOTE: The expected value for each cell needs to be at least 5 in order to use the Goodness-of-Fit, Independence and Homogeneity tests.

  • Homework

    • What are you ultimately interested in here (the value of the roll or the money you win)?
    • Over the long run of playing this game, what are your expected average winnings per game?
    • Based on numerical values, should you take the deal?
    • In words, what does the expected value in this example represent?
    • What does it mean that the values 0, 1, and 2 are not included for x in the PDF?

  • Plotting the Residuals

    • The residual plot illustrates how far away each of the values on the graph is from the expected value (the value on the line).
    • The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either.
    • The residual plot illustrates how far away each of the values on the graph is from the expected value (the value on the line).
    • So, based on the linear regression model, for a 2006 value of 415 drunk driving fatalities we would expect the number of drunk driving fatalities in 2009 to be lower than 377.
    • So, based on the linear regression model, for a 2006 value of 439 drunk driving fatalities we would expect the number of drunk driving fatalities for 2009 to be higher than 313.

  • Goodness-of-Fit Test

    • The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true.
    • These numbers are the expected (E) values.
    • To run the test, put the observed values (the data) into a first list and the expected values (the values you expect if the null hypothesis is true) into a second list.
    • To run the test, put the observed values (the data) into a first list and the expected values (the values you expect if the null hypothesis is true) into a second list.
    • To run the test, put the observed values (the data) into a first list and the expected values (the values you expect if the null hypothesis is true) into a second list.

  • Example: Test for Goodness of Fit

    • The Chi-square test for goodness of fit compares the expected and observed values to determine how well an experimenter's predictions fit the data.
    • These numbers are the expected ($E$) values.
    • The values in the table are the observed ($O$) values or data.
    • Now add (sum) the values of the last column.
    • To find the $p$-value, calculate $P$($\chi^2>3$).