Suppose that the distribution function of X is given by...

Suppose that the distribution function of $X$ is given by $$F(b)=\left\{\begin{array}{ll} 0 & b < 0 \\ \frac{b}{4} & 0 \leq b < 1 \\ \frac{1}{2}+\frac{b-1}{4} & 1 \leq b < 2 \\ \frac{11}{12} & 2 \leq b < 3 \\ 1 & 3 \leq b \end{array}\right.$$ (a) Find $P\{X=i\}, i=1,2,3$ (b) Find $P\left\{\frac{1}{2} < X < \frac{3}{2}\right\}$

Suppose that the distribution function of $X$ is given by
$$F(b)=\left\{\begin{array}{ll}
0 & b < 0 \\
\frac{b}{4} & 0 \leq b < 1 \\
\frac{1}{2}+\frac{b-1}{4} & 1 \leq b < 2 \\
\frac{11}{12} & 2 \leq b < 3 \\
1 & 3 \leq b
\end{array}\right.$$
(a) Find $P\{X=i\}, i=1,2,3$
(b) Find $P\left\{\frac{1}{2} < X < \frac{3}{2}\right\}$

Key Concepts

Probability Calculation over Intervals Using the CDF

Calculating the probability that a random variable lies within a certain interval using the CDF involves finding the difference between the CDF values at the endpoints of the interval. For intervals where the distribution is continuous, this calculation is straightforward. However, when the variable may also have discrete components, attention must be paid to whether the endpoints are included or excluded, as the CDF is right-continuous and jumps may occur at those points.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a key concept in probability theory that describes the probability that a random variable X is less than or equal to a particular value b. It is defined as F(b) = P(X ? b) and has properties such as being non-decreasing and right-continuous. The CDF provides a complete description of the distribution of a random variable, whether the variable is discrete, continuous, or a mixture of both.

Deriving the Probability Mass Function (PMF) from the CDF

For discrete random variables, the probability that X takes a specific value is given by the size of the jump in the CDF at that point. This is calculated as P(X = a) = F(a) - lim(b?a?) F(b). This concept is crucial when dealing with distributions that have both continuous and discrete components, as it allows one to extract the point masses directly from the CDF.

Piecewise-Defined Distribution Functions

A piecewise-defined distribution function represents a random variable whose behavior changes over different intervals. In such cases, the CDF is defined by different expressions depending on the range of the variable. This approach is common when the distribution includes both continuous regions and discrete jumps, requiring careful analysis within each specified interval to correctly interpret the overall distribution.

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