How should I understand the difference or relationship between binomial and Bernoulli distribution?

# probability – What is the difference and relationship between the binomial and Bernoulli distributions?

## The Question :

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*The Question Comments :*

## The Answer 1

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A Bernoulli random variable has two possible outcomes: $0$ or $1$. A binomial distribution is the sum of **independent** and **identically** distributed Bernoulli random variables.

So, for example, say I have a coin, and, when tossed, the probability it lands heads is $p$. So the probability that it lands tails is $1-p$ (there are no other possible outcomes for the coin toss). If the coin lands heads, you win one dollar. If the coin lands tails, you win nothing.

For a *single* coin toss, the probability you win one dollar is $p$. The random variable that represents your winnings after one coin toss is a Bernoulli random variable.

Now, if you toss the coin $5$ times, your winnings could be any whole number of dollars from zero dollars to five dollars, inclusive. The probability that you win five dollars is $p^5$, because each coin toss is independent of the others, and for each coin toss the probability of heads is $p$.

What is the probability that you win *exactly* three dollars in five tosses? That would require you to toss the coin five times, getting exactly three heads and two tails. This can be achieved with probability $\binom{5}{3} p^3 (1-p)^2$. And, in general, if there are $n$ Bernoulli trials, then the sum of those trials is binomially distributed with parameters $n$ and $p$.

Note that a binomial random variable with parameter $n = 1$ is equivalent to a Bernoulli random variable, i.e. there is only one trial.

## The Answer 2

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All Bernoulli distributions are binomial distributions, but most binomial distributions are not Bernoulli distributions.

If

$$

X=\begin{cases} 1 & \text{with probability }p, \\ 0 & \text{with probability }1-p, \end{cases}

$$

then the probability distribution of the random variable $X$ is a Bernoulli distribution.

If $X=X_1+\cdots+X_n$ and each of $X_1,\ldots,X_n$ has a Bernoulli distribution with the same value of $p$ and they are independent, then $X$ has a binomial distribution, and the possible values of $X$ are $\{0,1,2,3,\ldots,n\}$. If $n=1$ then that binomial distribution is a Bernoulli distribution.

## The Answer 3

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A Bernoulli distribution is a special case of binomial distribution. Specifically, when $n=1$ the binomial distribution becomes Bernoulli distribution.

## The Answer 4

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A Bernoulli random variable $X$ is a random variable that satisfies $P(X=1)=p$, $P(X=0)=1-p$. A canonical example is a coin flip which has $p=1/2$. In fact, you can think of a Bernoulli random variable is just a weighted coin, which comes up $1$ with some probability and $0$ otherwise. A binomial random variable with parameters $n,p$ is what you get when you count the number of $1$’s (successes) that come up in a string of $n$ independent Bernoulli random variables, each with parameter $p$. Another way to say this is that a binomial random variable is the sum of independent and identically distributed Bernoulli random variables.