Bernoulli distribution, binomial distribution, conceptual analysis

Bernoulli distribution

The Bernoulli distribution is a special case of the binomial distribution, which describes the probability distribution of only two outcomes in a single random experiment. where the probability of one outcome is $p$ , the probability of another outcome is$1-p$ . The probability mass function of the Bernoulli distribution is as follows:

$$f(k;p)=\{\begin{array}{ll}p& \text{if }k=1,\\ 1-p& \text{if }k=0.\end{array}$$

Among them, $k$ represents the outcome of the event,$p$ represents the probability of an event occurring.

A classic example of a Bernoulli distribution is a coin toss. In the process of tossing a coin, the probability of heads is $p$ , the probability of tails is$1-p$ . pphere$p$ is the parameter of the Bernoulli distribution.

binomial distribution

The binomial distribution is a kind of discrete probability distribution, which describes $In\; n$ independent repeated random trials, an event occurs$The\; probability\; distribution\; of\; k$ times. There are only two outcomes for each trial, success and failure. where the probability of success is$p$ , the probability of failure is$1-p$ . The probability mass function of the binomial distribution is as follows:

$$f(k;n,p)=\mathrm{Pr}(k;n,p)=\mathrm{Pr}(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},$$

Among them, $k$ represents the number of times the event occurred,$n$ represents the total number of trials,$p$ represents the probability of success for each trial.

A classic example of the binomial distribution is coin tossing. If we flip a coin once, then its outcome is a Bernoulli distribution. If we throw $n$ times the coin, then its result is a binomial distribution.

Discrimination between Bernoulli distribution and binomial distribution

The Bernoulli distribution is a special case of the binomial distribution, $n=1$ situation. Therefore, the binomial distribution is a superposition of multiple Bernoulli distributions. In practical applications, the Bernoulli distribution is usually used to describe the results of a single trial, while the binomial distribution is usually used to describe the results of multiple trials. For example, we can use the Bernoulli distribution to describe the outcome of a coin toss, and the binomial distribution to describe the outcome of tosses$n$ times coin, heads up$The\; result\; of\; k$ times.