If three dice are thrown once, find the distribution of the number of dice with an even number Three dice are thrown once and the random variable $X$ denotes the number of dice that brought an even number. Find:


*

*The probability function of $X$.

*The cumulative distribution function of $X$.

 A: It's the probability distribution of the number of successes in three independent trials with probability $1/2$ of success on each trial, so it's a binomial distribution.
A: The probability that a die shows an even number is $1/2$, so the problem is essentially the same as the problem of the probabilities of $0$ heads, $1$ head, $2$ heads, $3$ heads when we toss three fair coins. We get
$$f_X(0)=\frac{1}{8}; \qquad f_X(1)=\frac{3}{8}; \qquad f_X(2)=\frac{3}{8}; \qquad f_X(3)=\frac{1}{8}.$$ 
For the cumulative distribution function $F_X(x)$, recall that $F_X(x)=P(X\le x)$, and remember that $F_X(x)$ is defined for all real numbers $x$. In general, $F_X(x)$ is the "weight" up to and including $x$.
Then we can see that $F_X(x)=0$ for all $x<0$.  For $0 \le x <1$, we have $F_X(x)=\frac{1}{8}$. For $1 \le x <2$, we have $F_X(x)=\frac{4}{8}$. For $2 \le x <3$, we have $F_X(x)=\frac{7}{8}$. And finally, for $x \ge 1$, we have $F_X(x)=1$. The cumulative distribution function jumps upwards at $x=0$, $1$, $2$, and $3$. 
