The PMF is one way to describe the distribution of a discrete random variable. As we will see later on, PMF cannot be defined for continuous random variables. The cumulative distribution function (CDF) of a random variable is another method to describe the distribution of random variables. The advantage of the CDF is that it can be defined for any kind of random variable (discrete, continuous, and mixed). Show Definition Note that the subscript $X$ indicates that this is the CDF of the random variable $X$. Also, note that the CDF is defined for all $x \in \mathbb{R}$. Let us look at an example. Example I toss a coin twice. Let $X$ be the number of observed heads. Find the CDF of $X$.
In general, let $X$ be a discrete random variable with range $R_X=\{x_1,x_2,x_3,...\}$, such that $x_1 < x_2 < x_3 < ...$ Here, for simplicity, we assume that the range $R_X$ is bounded from below, i.e., $x_1$ is the smallest value in $R_X$. If this is not the case then $F_X(x)$ approaches zero as $x \rightarrow -\infty$ rather than hitting zero. Figure 3.4 shows the general form of the CDF, $F_X(x)$, for such a random variable. We see that the CDF is in the form of a staircase. In particular, note that the CDF starts at $0$; i.e.,$F_X(-\infty)=0$. Then, it jumps at each point in the range. In particular, the CDF stays flat between $x_k$ and $x_{k+1}$, so we can write $$F_X(x)=F_X(x_k), \textrm{ for }x_k \leq x < x_{k+1}.$$ The CDF jumps at each $x_k$. In particular, we can write $$F_X(x_k)-F_X(x_k-\epsilon)=P_X(x_k), \textrm{ For $\epsilon>0$ small enough.}$$ Thus, the CDF is always a non-decreasing function, i.e., if $y \geq x$ then $F_X(y)\geq F_X(x)$. Finally, the CDF approaches $1$ as $x$ becomes large. We can write $$\lim_{x \rightarrow \infty} F_X(x)=1.$$ Note that the CDF completely describes the distribution of a discrete random variable. In particular, we can find the PMF values by looking at the values of the jumps in the CDF function. Also, if we have the PMF, we can find the CDF from it. In particular, if $R_X=\{x_1,x_2,x_3,...\}$, we can write $$F_X(x)=\sum_{x_k \leq x} P_X(x_k).$$ Now, let us prove a useful formula. For all $a \leq b$, we have $$\hspace{50pt} P(a < X \leq b)=F_X(b)-F_X(a) \hspace{80pt} (3.1)$$ To see this, note that for $a \leq b$ we have $$P(X \leq b)=P(X \leq a) + P(a < X \leq b).$$ Thus, $$F_X(b)=F_X(a) + P(a < X \leq b).$$ Again, pay attention to the use of "$ < $" and "$\leq$" as they could make a difference in the case of discrete random variables. We will see later that Equation 3.1 is true for all types of random variables (discrete, continuous, and mixed). Note that the CDF gives us $P(X \leq x)$. To find $P(X < x)$, for a discrete random variable, we can simply write $$P(X < x)=P(X \leq x)-P(X=x)=F_X(x)-P_X(x).$$ Example
The print version of the book is available through Amazon here. What does the probability distribution of a discrete random variable tell you?The probability distribution of a discrete random variable X provides the possible value of the random variable along with their corresponding probabilities. A probability distribution can be in the form of a table, graph, or mathematical formula. Let's look at the earlier coin example to illustrate.
How to find probability distribution function of a random variable?Probability distribution for a discrete random variable.
The function f(x) p(x)= P(X=x) for each x within the range of X is called the probability distribution of X. It is often called the probability mass function for the discrete random variable X.
How to find probability distribution for discrete random variable?The mean μ of a discrete random variable X is a number that indicates the average value of X over numerous trials of the experiment. It is computed using the formula μ=∑xP(x).
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