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X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X is known to be 10/3. What would be the probability that X assumes at most the value 2?
  • a)
    16/81.
  • b)
    17/81.
  • c)
    47/243.
  • d)
    46/243
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X...
Given:

- X is a binomial variable
- 2 P(X = 2) = P(X = 3)
- Mean of X = 10/3

To find:

- Probability that X assumes at most the value 2

Solution:

1. Using the mean of X:

- Mean of X = np, where n is the number of trials and p is the probability of success in each trial
- np = 10/3
- p = (10/3)/n

2. Using the given relationship between P(X = 2) and P(X = 3):

- 2 P(X = 2) = P(X = 3)
- 2 (n C 2) p^2 (1-p)^(n-2) = (n C 3) p^3 (1-p)^(n-3)
- Simplifying and solving for p: p = 2/3

3. Using p = (10/3)/n:

- (10/3)/n = 2/3
- n = 5

4. Using the binomial distribution formula:

- P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
- P(X = k) = (n C k) p^k (1-p)^(n-k)
- P(X = 0) = (5 C 0) (2/3)^0 (1/3)^5 = 1/243
- P(X = 1) = (5 C 1) (2/3)^1 (1/3)^4 = 10/243
- P(X = 2) = (5 C 2) (2/3)^2 (1/3)^3 = 40/243
- P(X ≤ 2) = 1/243 + 10/243 + 40/243 = 17/81

Therefore, the probability that X assumes at most the value 2 is 17/81, which is option (B).
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X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X...
B
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X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X is known to be 10/3. What would be the probability that X assumes at most the value 2?a)16/81.b)17/81.c)47/243.d)46/243Correct answer is option 'B'. Can you explain this answer?
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