Mathematical Expectation
Mathematical expectation, also known as the expected value, is the summation or integration of a possible values from a random variable. It is also known as the product of the probability of an event occurring, denoted P(x), and the value corresponding with the actual observed occurrence of the event. The expected value is a useful property of any random variable. Usually notated as E(X), the expect value can be computed by the summation overall the distinct values that the random variable can take. The formula for mathematical expectation is E(X) = Σ (x1p1, x2p2, …, xnpn), where x is a random variable, p is its probability, and n is the number of possible values.
The mathematical expectation of an indicator variable can be zero if there is no occurrence of an event A, and the mathematical expectation of an indicator variable can be one if there is an occurrence of an event A. Thus, it is a useful tool to find the probability of event A.
Questions answered:
What is the expected number of coin flips for getting tails?
What is the expect number of coin flips for getting two tails in a row?
Properties and Assumptions:
The first property is that if X and Y are the two random variables, then the mathematical expectation of the sum of the two variables is equal to the sum of the mathematical expectation of X and the mathematical expectation of Y, provided that the mathematical expectation exists. In other words, E(X+Y)=E(X)+E(Y).
The second property is that the mathematical expectation of the product of the two random variables will be the product of the mathematical expectation of those two variables, provided that the two variables are independent in nature. In other words, E(XY)=E(X)E(Y).
The expectation of the product of n independent random variables equals the product of their expectations.
The expectation of a constant times a function equals the constant times the function’s expectation. The third property states E(a*f(X)) = a * E(f(X)) and E(a + f(X)) = a + E(f(X)), where a is constant.
The fourth property states that the expectation of a constant times a function plus another constant equals the sum. It equals the sum of the constant times the function’s expectation. In other words, E(aX+b)=aE(X)+b, where a and b are constants.
The fifth property states that the expectation of a linear combination equals the sum. It is the sum of constants times the expectations of the variables. In other words, E(∑aiXi)=∑ ai E(Xi). Here, ai, (i=1…n) are constants.
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