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in Probability
Sep 3, 2018

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Best answer

it can be done in different ways but i am using here hit and trial as that can be lengthy....

note: non decreasing(for a0,a1) means** a0<=a1**

consider it for **single symbol** : <**a0**>

**non decreasing sequence**= (0), (1),(2) =** 3sequence**

**total** sequence possible =** 3**( **ao** can contain 0,1,2)

**probability**(1symbol) = 3/3 =**1 **[email protected]

now consider it for** two symbol** sequence : **<a0,a1>**

so** non decreasing sequence** will be : (0,0) , (0,1) , (0,2) ,(1,1) ,(1,2) ,(2,2) =6

**total sequences from 3 symbols** = **3*3 **(a0 can take 3values = 0,1,2 also a1)

**probability**(2 symbols)= 6/9 =**2/3...**[email protected]

similarly for** three symbols** <a0,a1,a2>

**non decreasing sequence**= (0,0,0), (0,0,1), (0,0,2), (0,1,1), (0,1,2), (0,2,2) ,(1,1,1), (1,1,2) ,(1,2,2), (2,2,2)=**10sequences**

**total sequences = 3*3*3 **

**probability**(3 symbols) = **10/27** [email protected]

checking option A for above case :

(n=1) = (1+3*1 +2)/(2 *3) = 6/6 =1 [email protected] proved

(n=2) = (2*2+ 3*2 +2) / (2*3*3) = 12/18 = 2/[email protected] proved

(n=3) = (3*3 +3*3 +2)/(2*3*3*3)=20/54=10/[email protected] proved.

so answer is **option A**.

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@arvin

Actually , for homogeneous part ,equation becomes "r^2 -2r+1".

Find its roots, then find solution for homogeneous part.

Then solve for non homogeneous part.

A bit lengthy process.

https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_recurrence_relation.htm

This will help or there are YouTube vid.

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