Question

1. In a batch of 10 items, we wish to extract a sample of 3 without replacement. How many

different samples can we extract?
2. Let X and Y be independent random variables. Suppose the respective expected values
are E(X) = 8 and E(Y) = 3 and the respective variances are V(X) = 9 and V(Y) = 6. Let Z be
defined as Z = 2X - 3Y +5. Based on these data, the value of E(Z) is_ and the value
of V(Z) is
3. The ratio of milk water in 55 liters of adulterated milk is 7: 4. How much water must be
added to make the mixture in a ratio of 7:6?
a) 5 liters
b) 10 liters
c) 15 liters d) None of these​

Answer 1

1) 120

2) E (Z) = 12 and Variance of Z = 90

a) 5 liters

Explanation:

1. We can find this by suing combinations.

Here n= 10 and r= 3 so n C r

= 10 C 3= 120

2. E(X) = 8 and E(Y) = 3

Z = 2X - 3Y +5

E(Z ) = 2 E (X) - 3(E)(Y) +5 ( applying property for mean)

= 2(8) - 3(3)+ 5 = 16+5-9= 21-9= 12

V(X) = 9 and V(Y) = 6.

V(Z) = E(Z )²- V(X) *V(Y) (applying property for Variance for two variables )

= 144- 54= 90

3. 55 liters contain adulterated milk in 7: 4.

So it contains 4/ 11*55= 20 liters of water

But we want to make it a ratio of 7:6

the water will be 6/13 *55= 25.38 when rounded gives 25 liters of water

So 25- 20 = 5 liters must be added to make it a ratio of 7:6