Learning of the basic Info regarding Sigma Notation & Expected Value

Why Do We Need Symbols & Notations?

Language helps to put things in order and which were previously unable to address. It was needless to say humans use language to express their emotions and thoughts, thus providing a sustainable medium to convey ideas and sending the messages. There is a famous adage that states that Mathematics is the language of the universe; this ostensibly refers to the fact that Mathematics has provided the notations and symbols which, in the future, become the blueprint to demystify the universal problems. Be it Geometry, Algebra, or Calculus, the symbols have simplified the complex calculations and paved the path for innovations.

What is Sigma Notation?

Sigma notation? is widely used to denote the summation in a process. It originates from the Greek alphabet, it is the capital form of?. This notation deals with the various addition operators involving a sequence of numbers. Any number or set of numbers written immediately after sigma notation will be the subject of summation. For example, if a set ‘n’ contains the list of first 10 natural numbers then ‘? n’ will produce the sum of the first 10 natural numbers.

It can be customized for specific numbers as well. You can describe the set of numbers within the symbol if you need to have a sum of specific numbers.

Similarly, if the suffix part or a parameter bears any change, the subsequent summation will be changed. E.g.,

The suffix expression is called summation; it appears immediately after the sigma notation. Whereas the numbers 1 to 4 or 1 to 2 are called limits to which numbers need to be added in a given expression.

What is the Expected Value?

It goes without saying that we perform calculations to reach certainty in daily life problems. It assists in making definite decisions rather than making random guesses. A simple way to think about the expected value is to decide whether you should purchase a home or not in given conditions of interest rate, mortgage, and location.

The theory of Probability gives you an idea for how much there is a possibility that your event would occur in a particular way. On the other side, the Expected Value (EV) gives an estimated value after the completion of the process. Generally, an expected value is the theoretical mean obtained after the sequence of repetitive numerical experiments. It does not necessarily guarantee accuracy but it shows the tendency of the numerical experiment.

Expected value can be seen in two ways, i.e., for discrete random variable and continuous random variable. In the case of a discrete random variable, the expected value can be categorized as the arithmetic mean of the set of given values to their probability of occurrence. The expected value for the discrete random variable is,

E(X) =?sx P(X=x)

Where X is the set of discrete random variable with the probability mass functions of,

P(X=1) =?

P(X=2) =?

P(X=3) = ½

Putting them in the above formula will give,

E(X) = 1(1/8) + 2(3/8) + 3(1/2)

= 2.375

Now, when it comes to the continuous random variable, the formula for EV will be,

If the continuous function f(x) contains the value of x for the interval 0? x? 1, 2-x for interval 1

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