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I am a beginner in statistics and probability study. Today I came across a new topic called the Mathematical Expectation. The book I am following says expectation is the arithmetic mean of any probability distribution. But it defines expectation as the product of some data and the probability of it. How can these two be same? How can the probability times the data be the average of whole distribution?? Thanks in advance.. |
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Informally, a probability distribution defines the relative frequency of outcomes of a random variable - the expected value can be thought of as a weighted average of those outcomes (weighted by the relative frequency). Similarly, the expected value can be thought of as the arithmetic mean of a set of numbers generated in exact proportion to their probability of occurring (in the case of a continuous random variable this isn't exactly true since specific values have probability $0$). The connection between the expected value the arithmetic mean is most clear with a discrete random variable, where the expected value is $$ E(X) = \sum_{S} x P(X=x) $$ where $S$ is the sample space. As an example, suppose you have a discrete random variable $X$ such that: $$ X = \begin{cases} 1 & \mbox{with probability } 1/8 \\ 2 & \mbox{with probability } 3/8 \\ 3 & \mbox{with probability } 1/2 \end{cases} $$ That is, the probability mass function is $P(X=1)=1/8$, $P(X=2)=3/8$, and $P(X=3)=1/2$. Using the formula above, the expected value is $$ E(X) = 1\cdot (1/8) + 2 \cdot (3/8) + 3 \cdot (1/2) = 2.375 $$ Now consider numbers generated with frequencies exactly proportional to the probability mass function - for example, the set of numbers $\{1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3\}$ - two $1$s, six $2$s and eight $3$s. Now take the arithmetic mean of these numbers: $$ \frac{1+1+2+2+2+2+2+2+3+3+3+3+3+3+3+3}{16} = 2.375 $$ and you can see it's exactly equal to the expected value. |
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The expectation is the average value or mean of a random variable not a probability distribution. As such it is for discrete random variables the weighted average of the values the random variable takes on where the weighting is according to the relative frequency of occurrence of those individual values. For an absolutely continuous random variable it is the integral of values x multiplied by the probability density. Observed data can be viewed as the values of a collection of independent identically distributed random variables. The sample mean (or sample expectation) is defined as the expectation of the data with respect to the empirical distribution for the observed data. This makes it simply the arithmetic average of the data. |
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