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MatroZZZ [7]
3 years ago
8

A fair coin is tossed repeatedly with results Y0, Y1, Y2, . . . that are 0 or 1 with probability 1/2 each. For n ≥ 1 let Xn = Yn

+ Yn−1 be the number of 1’s in the (n − 1)th and nth tosses. Is Xn a Markov chain?
Mathematics
1 answer:
Gekata [30.6K]3 years ago
6 0

Answer:

False. See te explanation an counter example below.

Step-by-step explanation:

For this case we need to find:

P(X_{n+1} = | X_n =i, X_{n-1}=i') =P(X_{n+1}=j |X_n =i) for all i,i',j and for X_n in the Markov Chain assumed. If we proof this then we have a Markov Chain

For example if we assume that j=2, i=1, i'=0 then we have this:

P(X_{n+1} = | X_n =i, X_{n-1}=i') =\frac{1}{2}

Because we can only have j=2, i=1, i'=0 if we have this:

Y_{n+1}=1 , Y_n= 1, Y_{n-1}=0, Y_{n-2}=0, from definition given X_n = Y_n + Y_{n-1}

With i=1, i'=0 we have that Y_n =1 , Y_{n-1}=0, Y_{n-2}=0

So based on these conditions Y_{n+1} would be 1 with probability 1/2 from the definition.

If we find a counter example when the probability is not satisfied we can proof that we don't have a Markov Chain.

Let's assume that j=2, i=1, i'=2 for this case in order to satisfy the definition then Y_n =0, Y_{n-1}=1, Y_{n-2}=1

But on this case that means X_{n+1}\neq 2 and on this case the probability P(X_{n+1}=j| X_n =i, X_{n-1}=i')= 0, so we have a counter example and we have that:

P(X_{n+1} =j| X_n =i, X_{n-1}=i') \neq P(X_{n+1} =j | X_n =i) for all i,i', j so then we can conclude that we don't have a Markov chain for this case.

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One gram of a certain health food contains 7 percent of the minimum daily requirement of vitamin E and 3 percent of the minimum
Liula [17]

Answer:

Approximately 34 grams of the healthy food

Step-by-step explanation:

For know the minimum value to be eaten daily to provide the requirement of both vitamins, is necessary to calculate the minimum value for every vitamin, so:

  • vitamin E: We are going to use the rule of three in which 1 gram have 7% of the minimum daily requirement, then how many grams are going to be be the 100% of the daily requirement:

1 gram --------------7%

X grams -----------100%

X=\frac{1 gram* 100}{ 7 } = 14.285 grams

So, it is necessary approximately 15 grams of healthy food to complete th 100% of the minimum daily requirement of vitamin E.

  • vitamin A: We are going to use the rule of three in which 1 gram have 3% of the minimum daily requirement, then how many grams are going to be be the 100% of the daily requirement:

1 gram --------------3%

X grams -----------100%

X=\frac{1 gram* 100}{ 3 } = 33.33 grams

So, it is necessary approximately 34 grams of healthy food to complete th 100% of the minimum daily requirement of vitamin E.

Finally for satisfy with both minimum daily requirement, we need to eat at least 34 grams of healthy food, because it is the maximum between the two X values.

3 0
3 years ago
In the problem 3 x 5 = 15, the 3 and the 5 are called WHAT IS IT Called
atroni [7]

Answer:

factors ?

Step-by-step explanation:

7 0
3 years ago
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Yuri [45]
Let's say that the diameter was 1/2 mile. So the diameter is 1/2,  then you need to find the circumference, aka perimeter of the circle. So 1/2 times 3.14 = 1.57 miles So then you would do 1.57/2 = .785 of an hour If you round to nearest tenth The answer is .8 of an hour.
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29. The function f(t) =-16t^2 + 35 gives the height, f (in feet) of a ball t seconds after being dropped from a height of 35 fee
Nostrana [21]

The time taken for the ball to fall a distance of 10feet is 1.25secs

Given the function modeled by the height expressed as:

f(t) =-16t² + 35

In order to determine the time taken by the all to fall a distance of 10 feet, hence;

f(t) = 10

10 = -16t² + 35

-16t² = 10 - 35

-16t² = -25

-t² = -25/16

t² = 25/16

t = √25/16

t = 5/4

t = 1.25secs

Hence the time taken for the ball to fall a distance of 10feet is 1.25secs

Learn more here: brainly.com/question/13881652

8 0
3 years ago
Find the value for d<br> 2d - 5 = 17
Ann [662]
2d-5=17

2d=17+5

2d=22
— —
2 2

d=11
4 0
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