Page 37 - 4660

Arithmetical Operations with DRV

We may also compute the probability that X lies between two limits, l 1 and l 2 (l 1 < l 2 ); this is
given by
∑
P(l 1 < X ≤ l 2 ) = f(x i ) = F(l 2 ) − F(l 1 ), (5.4)
l 1 <x i ≤l 2
i.e. it is the sum of all the probabilities for which x i lies within the relevant interval.

Example 5.2. A bag contains seven red balls and three white balls. Three balls
are drawn at random and not replaced. Find the probability function for the number
of red balls drawn. ,

Solution. Let X be the number of red balls drawn. Then

3 2 1 1
P(X = 0) = f(0) = × × = ,
10 9 8 120
3 2 7 7
P(X = 1) = f(1) = × × × 3 = ,
10 9 8 40
3 7 6 21
P(X = 2) = f(2) = × × × 3 = ,
10 9 8 40
7 6 5 7
P(X = 3) = f(3) = × × = .
10 9 8 24
3
∑
It should be noted that f(i) = 1, as expected.
i=0

Arithmetical Operations with DRV

Definition 5.4. Two DRV are called independent if the law of distribution of
one of them does not depend on what possible values were accepted by other
DRV. ✓
Let independent DRV X and Y be denoted by the distribution tables
X x 1 x 2 … x m
P p 1 p 2 … p m
Y y 1 y 2 … y n
P q 1 q 2 … q n

Definition 5.5. By the sum of two independent DRV X and Y we mean a DRV
which accepts all possible values x i + y j with probabilities p ij defined by the
product theorem. ✓
In the case of independent variables we have

p ij = p i q j . (5.5)

Definition 5.6. By the product of two independent DRV X and Y we mean a DRV
which accepts all possible values x i y j with probabilities p ij defined by the product
theorem. ✓
Definition 5.7. By the k-th degree of a DRV X we mean a DRV which takes
k
values x with the same probabilities p i . ✓
i
Definition 5.8. By a product of a DRV X and a constant A we mean a DRV
which accepts values Ax i with the same probabilities. ✓

2
Remark 5.1. For DRV X · X ̸= X , X + X ̸= 2X.

37