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For n independent trials in Binomial distribution the sum of the powers of p and q is always n , whatever be the no. of success.
- a)True
- b)false
- c)both
- d)none
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
For n independent trials in Binomial distribution the sum of the power...
Binomial Distribution and Sum of Powers of p and q
Binomial distribution is a type of probability distribution that describes the probability of a certain number of successes in a fixed number of independent trials. The distribution is characterized by two parameters - the probability of success (p) and the probability of failure (q), where q = 1 - p.
The sum of powers of p and q is always equal to n, where n is the number of independent trials. This property holds true for all values of the number of successes.
Proof:
Let X be a random variable that follows a binomial distribution with parameters n and p. Then, the probability mass function of X is given by:
P(X = k) = nCk p^k q^(n-k)
where nCk is the binomial coefficient.
The sum of powers of p and q can be expressed as:
p^n + q^n = (p+q)^n
Using the binomial theorem, we can expand (p+q)^n as:
(p+q)^n = ∑_(k=0)^n nCk p^k q^(n-k)
Comparing this with the probability mass function of X, we can see that the sum of powers of p and q is equal to the sum of the probabilities of all possible outcomes of X. Since the probabilities of all possible outcomes of X add up to 1, we have:
p^n + q^n = (p+q)^n = 1^n = 1
Therefore, the sum of powers of p and q is always equal to n, regardless of the number of successes.
Conclusion:
Hence, we can conclude that the statement "For n independent trials in Binomial distribution the sum of the powers of p and q is always n , whatever be the no. of success" is true.
Binomial distribution is a type of probability distribution that describes the probability of a certain number of successes in a fixed number of independent trials. The distribution is characterized by two parameters - the probability of success (p) and the probability of failure (q), where q = 1 - p.
The sum of powers of p and q is always equal to n, where n is the number of independent trials. This property holds true for all values of the number of successes.
Proof:
Let X be a random variable that follows a binomial distribution with parameters n and p. Then, the probability mass function of X is given by:
P(X = k) = nCk p^k q^(n-k)
where nCk is the binomial coefficient.
The sum of powers of p and q can be expressed as:
p^n + q^n = (p+q)^n
Using the binomial theorem, we can expand (p+q)^n as:
(p+q)^n = ∑_(k=0)^n nCk p^k q^(n-k)
Comparing this with the probability mass function of X, we can see that the sum of powers of p and q is equal to the sum of the probabilities of all possible outcomes of X. Since the probabilities of all possible outcomes of X add up to 1, we have:
p^n + q^n = (p+q)^n = 1^n = 1
Therefore, the sum of powers of p and q is always equal to n, regardless of the number of successes.
Conclusion:
Hence, we can conclude that the statement "For n independent trials in Binomial distribution the sum of the powers of p and q is always n , whatever be the no. of success" is true.
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For n independent trials in Binomial distribution the sum of the power...
The powers of p and q is x and n-x
sum = x + n - x = n
sum = x + n - x = n
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