# What did you learn from binomial and Hyper- geometric distributions?

What did you learn from binomial and Hyper- geometric distributions? Write a brief note of five lines on these distributions.

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Foreckije
Step 1
BINOMIAL DISTRIBUTION
$$\displaystyle\Rightarrow$$ It is a discrete probability distribution which gives the theoretical probabilities.
$$\displaystyle\Rightarrow$$ Binomial distribution allow us to deal with circumstances in which the outcomes belong to two relevant categories such as success or failure.
$$\displaystyle\Rightarrow$$ It depends on the parameter p or q i.e. the probability of success or failure and n (i.e. the number of trials). The parameter n is always a positive integer.
$$\displaystyle\Rightarrow$$ The distribution will be symmetrical if $$\displaystyle{p}={q}$$. It is skew-symmetric or asymmetric if $$\displaystyle{p}\ne{q}$$.
$$\displaystyle\Rightarrow$$ The possibility of outcome of any trial does not change and is independent of the results of previous trials.
Formula
$$\displaystyle{P}{\left({x}\right)}=^{{{n}}}{C}_{{{x}}}\cdot{p}^{{{x}}}\cdot{q}^{{{n}-{x}}}$$ where $$\displaystyle{x}\Rightarrow$$ number of successes
Step 2
HYPER GEOMMETRIC DISTRIBUTION
$$\displaystyle\Rightarrow$$ A hypergeometric distribution has a specified number of dependent trials having two possible outcomes, success or failure.
$$\displaystyle\Rightarrow$$ The random variable is the number of successful outcomes in the specified number of trials.
$$\displaystyle\Rightarrow$$ In statistics and probability theory, the hypergeometric distribution is a discrete probability distribution that describes the probability of successes in draws without replacement.
$$\displaystyle\Rightarrow$$ The probability of a success is not the same on each trial without replacement, thus events are not independent.
Formula
$$\displaystyle{P}{\left({x}\right)}={\frac{{{s}{C}_{{{k}}}\cdot^{{{N}-{S}{C}_{{{N}-{S}}}}}}}{{{N}{C}_{{{n}}}}}}$$
where $$\displaystyle{S}\Rightarrow$$ successes from population
$$\displaystyle{N}\Rightarrow$$ population size
$$\displaystyle{n}\Rightarrow$$ sample size
$$\displaystyle{k}\Rightarrow$$ successes from sample
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ramirezhereva

Step 1
Consider a random experiment involving n independent trials, such that the outcome of each trial can be classified as either a “success” or a “failure”. The numerical value “1” is assigned to each success and “0” is assigned to each failure.
Moreover, the probability of getting a success in each trial, p, remains a constant for all the n trials. Denote the probability of failure as q. As success and failure are mutually exclusive, $$\displaystyle{q}={1}-{p}$$.
Let the random variable X denote the number of successes obtained from the n trials. Thus, X can take any of the values $$\displaystyle{0},\ {1},\ {2},с\dot{{s}},\ {n}.$$
Then, the probability distribution of X is a Binomial distribution with parameters (n, p) and the probability mass function (pmf) of X, that is, of a Binomial random variable, is given as:
$f(x)=\begin{cases}(\begin{array}{c}n\\ x\end{array})p^{x}q^{n-x} & x=0,\ 1,\dots\end{cases} Step 2 Hyper geometric distribution: A hyper geometric distribution is a discrete probability distribution that determines the probability of getting k successes in n draws (without replacement) from a finite population of size N that contains exactly K success states. Denote the total number of successes as k, Denote the total number of objects that are drawn without replacement as n, Denote the population size as N, Denote the total number of success states in the population as K. The probability distribution of k is a hyper geometric distribution with parameters (N, K, n) and the probability mass function (pmf) of k is given as: \[P(k)=\frac{(\begin{array}{c}K\\ k\end{array})\times(\begin{array}{c}N-K\\ n-k\end{array})}{(\begin{array}{c}N\\ n\end{array})}$

nick1337
Binomial distribution  :
1. Trials are independent
2. Occurences are classified into 2 categories namely success and failure
3. Probability of success is a constant in every single trial
4. Trials are WR
5. Population is finite .
Hypergeometric distribution :
1.  Trials are WOR
2. Trials are dependent .
3. Probability of success changesw in every trial.
4. When population size is large , binomial becomes approximately equal to hypergeometric .
5. Population is infinite