Expected Value of a Continuous Random Variable?

As we all know that if range of Random Variable is uncountable or infinite, then this type of random variables are called as a continuous random variable like we have ‘X’ as a random variable, then range of this random variable ‘X’ is uncountable or infinite for all kind of physical measurements like size, age, flow, volume, area in continuous random variable. Now we will discuss how to calculate expected value of a continuous random variable:
We use following formula for expected value of a continuous random variable -
Expected value of a continuous random variable Y = E(Y) = ∫_-∞^∞ y. f(y). dy,
Now we will take an example to understand the process of finding the expected value from continuous random variable:
Example: Find the expected value of following continuous random variable function -
Y = f(y) = (3/2) y² + y, where 0 <= y <= 1?
Solution: We will use above expected value formula for continuous random variable -
E(Y) = ∫_-∞^∞ y. f(y) . dy,
Because here limits of this continuous random variable is given in question means
0 <= y <= 1 so, lower limit of ‘y’ is 0 and upper limit of ‘y’ is 1.
= ∫₀¹ y . [(3 / 2) y² + y] . dy,
= ∫₀¹ [(3 / 2) y³ + y²] . dy,
= ∫₀¹ [(3 / 2) y³ . dy + ∫₀¹ y² . dy,
= (3 / 2) . [y⁴ / 4]₀¹ + [y³ / 3]₀¹,
= (3 / 8) . [1 – 0] + 1 / 3. [1 – 0],
= 3 / 8 + 1 / 3.
= 17/24.
So, expected value of this continuous random variable is 17/24.
Therefore it is very easy way to find expected value of continuous random variable from above Integration method, but here limit part is important.