The general Probability Theory that is read by us in the school and colleges doesn't include the Probability in Calculus but rather is a subject of discussion in higher class Math. The probability in calculus includes the study of Statistics and probability but it is rather based upon the continuous and discrete aspect.
The probability theory in calculus uses the powerful Algebra, graphs and statistics. Since the probability is always non- negative and on the basis on that in this part of calculus we have two conclusions:
p i > = 0
∑ i p i = 1.
The discrete section of Probability Distribution is generally presented as either a table or a formula. Let us take an example to understand the concept of probability, the example is shown below-
Example: Let Y be the Random Variable with uniform distribution on interval [ 0 , 1 ], often denoted Y ∼ unif [ 0 , 1 ] . Then
f ( y ) = 1 if 0 ≤ y ≤ 1
Now we have to Calculate The Probability that Y falls between .3 and .7, that is
P ( .3 < Y < .7 ).
Here we will integrate pdf over the values specified-
P ( .3 < Y < .7 ) = ∫.3.7 f ( y ) dy = ∫.3.7 1 . dy = .7 - .3 = .4
The Mean and variance can be found as-
E [ Y ] = ∫-∞∞ y . f (y) .dy = ∫01 1 . dy = 1 / 2,
Var ( Y ) = E [ Y 2 ] - ( E [ Y ] )2 = 1 / 3 – 1 / 4 = 1 / 12.