Expectation of Random Variable
An “expectation” or the “expected value” of a random variable is the value that you would expect the outcome of some experiment to be on average. The expectation is denoted by E(X).
The expectation of a random variable can be computed depending on the type of random variable you have.
For a Discrete Random Variable,
E(X) = ∑x × P(X = x)
For a Continuous Random Variable,
E(X) = ∫x × f(x)
Where,
- The limits of integration are –∞ to + ∞ and
- f(x) is the probability density function
Example: What is the expectation when a standard unbiased die is rolled?
Solution:
Rolling a fair die has 6 possible outcomes : 1, 2, 3, 4, 5, 6 each with an equal probability of 1/6
Let X indicate the outcome of the experiment
Thus P(X=1) = 1/6
⇒ P(X=2) = 1/6
⇒ P(X=3) = 1/6
⇒ P(X=4) = 1/6
⇒ P(X=5) = 1/6
⇒ P(X=6) = 1/6
Thus, E(X) = ∑ x × P(X=x)
⇒ E(X) = 1× (1/6) + 2 × (1/6) + 3 × (1/6) + 4 × (1/6) + 5 × (1/6) + 6 × (1/6)
⇒ E(X) = 7/2 = 3.5
This expected value kind of intuitively makes sense as well because 3.5 is in halfway in between the possible values the die can
take and thus this is the value that you could expect.
Properties of Expectation
Some properties of Expectation are as follows:
- In general, for any function f(x) , the expectation is E[f(x)] = ∑ f(x) × P(X = x)
- If k is a constant then E(k) = k
- If k is a constant and f(x) is a function of x then E[k f(x)] = k E[f(x)]
- Let c1 and c2 be constants and u1 and u2 are functions of x then E = c1E[u1(x)] + c2E[u2(x)]
Example: Given E(X) = 4 and E(X2) = 6 find out the value of E(3X2 – 4X + 2)
Solution:
Using the various properties of expectation listed above , we get
E(3X2 – 4X + 2) = 3 × E(X2) – 4 × E(X) + E(2)
= 3 × 6 – 4 × 4 + 2
= 4
Thus, E(3X2 – 4X + 2) = 4
Read More,
Discrete Random Variable
Discrete Random Variables are an essential concept in probability theory and statistics. Discrete Random Variables play a crucial role in modelling real-world phenomena, from the number of customers who visit a store each day to the number of defective items in a production line. Understanding discrete random variables is essential for making informed decisions in various fields, such as finance, engineering, and healthcare. In this article, we’ll delve into the fundamentals of discrete random variables, including their definition, probability mass function, expected value, and variance. By the end of this article, you’ll have a solid understanding of discrete random variables and how to use them to make better decisions.