Bernoulli Distribution

Topics: Probability - Distribution Function

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(definition)

Let $X$ be a random variable that handles a single Bernoulli trial as follows:

$$X(\omega )=\{\begin{array}{ll}0& \text{if }\omega =\text{failure}\\ 1& \text{if }\omega =\text{success}\end{array}$$

Let $p$ be the probability of success.

In such a case, we say that $X$ has a Bernoulli distribution and write $X\sim \mathrm{Ber}(p)$.

Single vs. Multiple Trials

If $X$ handles a single Bernoulli trial, then it has a Bernoulli distribution.

On the other hand, when it handles multiple Bernoulli trials (of the same nature), it has a binomial distribution.

Density Function §

(theorem)

The density function of such an $X$ is given by:

$$f_X(x)=\{\begin{array}{ll}p& \text{if }x=1\\ 1-p& \text{if }x=0\\ 0& \text{otherwise}\end{array}$$

Distribution Function §

(theorem)

As for the distribution function of such an $X$:

$$F_X(X)=\{\begin{array}{ll}0& \text{if }x<0\\ 1-p& \text{if }0\le x<1\\ 1& \text{if }1\le x\end{array}$$

Expected Value §

(theorem)

When it comes to the expected value of such an $X$:

$$\mathbb{E}[X]=p$$

Variance §

(theorem)

For the variance of such an $X$:

$$\mathrm{Var}(X)=p(1-p)$$

Moment-Generating Function §

(theorem)

Such an $X$ has its moment-generating function given by:

$$M_X(t)=e^{t}p+(1-p)$$

Characteristic Function §

(theorem)

Finally, the characteristic function of $X$ is given by:

$${\phi }_X(t)=e^{it}p+(1-p)$$