Suppose that the cumulative distribution function of the...

Suppose that the cumulative distribution function of the random variable $X$ is
$$
F(x)=\left\{\begin{array}{lr}
0 & x<0 \\
0.25 x & 0 \leq x<5 \\
1 & 5 \leq x
\end{array}\right.
$$
Determine the following:
(a) $P(X<2.8)$
(b) $P(X>1.5)$
(c) $P(X<-2)$
(d) $P(X>6)$

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Key Concepts

Cumulative Distribution Function (CDF)

A Cumulative Distribution Function (CDF) for a random variable provides the probability that the variable takes on a value less than or equal to a given number. It is defined for all real numbers and is a non-decreasing function that approaches 0 as x starts from the lower end of the domain and reaches 1 at the upper end. Its behavior summarizes the entire distribution of a random variable.

Computing Probabilities Using the CDF

Using the CDF, probabilities for intervals or specific events can be computed by evaluating the CDF at the endpoints. For example, to find the probability that a random variable is less than a certain value, one directly uses the CDF at that value. Similarly, for events such as exceeding a particular value, the complementary probability (1 minus the CDF value) is used. This method is particularly useful when the CDF is defined piecewise, as it allows for straightforward computation of probabilities across different segments of the distribution.

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