Derivation of the probability distribution function of a binomial distribution

In summary, the formula for P(X=r) in a binomial distribution B(n,p) is ^nC_r p^r q^{n-r}, where n is the total number of Bernoulli experiments, p is the probability of success, and q is the probability of failure. This can be easily derived using basic combinatorics principles, such as choosing r from n. The number of combinations can be calculated by taking the factorial of the total number of elements and dividing it by the factorial of the number of similar elements.
  • #1
misogynisticfeminist
370
0
Is there a way to derive

[tex] P (X=r) =^nC_r p^r q^{n-r} , r= 0, 1, 2,..., n [/tex]

where [tex] X: B(n,p) [/tex]

where n is the total number of bernoulli experiments,

p the probability of success

q, the probability of failure.
 
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  • #2
Yes, just think about it, it's just simple combinatorics (you omitted to mention independent trials, which is, I'm sure important), and no harder than working out how to choose r from n (in fact it is the same).
 
  • #3
This is not a problem as matt goes into. Suppose we have XXX and YY,then how many ways can combinations occur? Well there are five elements in 5! ways, but 3 of them are similar and the other two are similar, so it's 5!/(3!2!)=10 distinct ways. YYXXX, YXYXX, YXXYX, YXXXY, XYXXY, XXYXY, XXXYY, XYYXX, XXYYX, XYXYX.
 

1. What is a binomial distribution?

A binomial distribution is a probability distribution that describes the likelihood of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (usually referred to as success and failure).

2. How is the probability distribution function of a binomial distribution derived?

The probability distribution function of a binomial distribution is derived using the binomial theorem and the concept of combinations. It involves calculating the probability of obtaining a specific number of successes in a given number of trials, taking into account the probability of success and failure in each trial.

3. What is the formula for the probability distribution function of a binomial distribution?

The formula for the probability distribution function of a binomial distribution is P(x) = nCx * p^x * (1-p)^(n-x), where n is the number of trials, p is the probability of success in each trial, and x is the number of successes.

4. Can the probability distribution function of a binomial distribution be used to find the probability of obtaining a range of values?

Yes, the probability distribution function of a binomial distribution can be used to find the probability of obtaining a range of values by summing the individual probabilities of each possible outcome within that range. For example, to find the probability of obtaining between 3 and 5 successes in 10 trials, you would calculate P(3) + P(4) + P(5).

5. What are some real-life applications of the binomial distribution?

The binomial distribution is commonly used in fields such as statistics, psychology, and biology to describe and analyze the results of experiments or studies with binary outcomes. It is also used in quality control and market research to determine the likelihood of success or failure in a given situation.

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