From this discussion, the notion of expected value was introduced.

( b ) greater than expected growth in the value of the guaranteed benefits

This relationship can be used to translate properties of expected values into properties of probabilities, e.g.

The bias of an estimator is the difference between an estimator's expectations and the true value of the parameter being estimated.

For example, suppose random variable X takes values 1, 2, 3, 4, ..., with respective probabilities , , , , ..., where is a normalizing constant that ensures the probabilities sum up to one.

For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique.

One example sequence of ten rolls of the is 2, 3, 1, 2, 5, 6, 2, 2, 2, 6, which has the average of 3.1, with the distance of 0.4 from the expected value of 3.5.

Moments and cumulants The expected value of a geometrically distributed random variable X is 1 p and the variance is (1 p ) p 2 formula_21Similarly, the expected value of the geometrically distributed random variable Y X 1 (where Y corresponds to the pmf listed in the right column) is q p (1 p ) p , and its variance is (1 p ) p 2 formula_22Let μ (1 p ) p be the expected value of Y .

This is captured by the mathematical concept of expected value of a random variable, denoted E X , and also called the first moment.

The pre announcement helped to stabilise the ERM for the remainder of the year by reducing uncertainties regarding the changeover to the single currency .

The error in the a posteriori state estimation is formula_46We seek to minimize the expected value of the square of the magnitude of this vector, formula_47.

But we were expected to have doubts we're expected to have fears.

I would've waited.

Another billion is expected.

It went just as we expected.

Thanks for waiting.

You should've waited.

Everything went as expected.

You should have waited.

You should have waited.

The formal definition subsumes both of these and also works for distributions which are neither discrete nor continuous the expected value of a random variable is the integral of the random variable with respect to its probability measure.

Never keep a lady waiting.

I've been waiting for that question.

That should've been expected.

You should've expected us.

I should've waited for Tom.

Now, if you'd waited, if you'd waited for one more year, you would have heard this.

As suspected, they provoked CBS.

I should've waited for Tom.

The outcome was not what had been expected.

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This is somewhat higher than expected a few months ago and appears to be related , in particular , to upward pressure in energy prices .

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Something you didn't expect that caps what you did expect.

It was the opposite of what was expected.

I have waited for your salvation, Yahweh.

We spent more money than was expected.

I should've waited another three months.

I was expected here before ten.

I should've expected that to happen.

It's everything we could have hoped for.

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In other words, it all went exactly as expected.

( c ) the expected return on the plan assets