3.1 .13 x 1.25 1.5 1.75 2 2.25...

$$ \begin{array}{lcccccc} 3.1 .13 & x & 1.25 & 1.5 & 1.75 & 2 & 2.25 \\ & f(x) & 0.2 & 0.4 & 0.1 & 0.2 & 0.1 \\ \text { a. } & P(X \geq 2) & \text { b. } P(X<1.65) & & \\ \text { c. } & P(X=1.5) & \text { d. } P(X<1.3 \text { or } X>2.1) \end{array} $$

Key Concepts

Event Probability Calculation

Calculating the probability of an event involving a discrete random variable often involves summing the probabilities of individual outcomes that satisfy the event's conditions. This includes adding probabilities for mutually exclusive events or using the complement rule when it is more efficient to calculate the probability of the opposite event.

Cumulative Distribution Function (CDF)

The cumulative distribution function describes the probability that a discrete random variable will take a value less than or equal to a certain number. It is obtained by summing the probabilities from the PMF for all values up to the desired point, and it is useful for calculating probabilities involving inequalities.

Discrete Random Variable

A discrete random variable is a variable that can take on a countable number of distinct outcomes. In probability theory, it is used to model situations where outcomes are distinct and separate, as opposed to a continuous random variable which can take any value within an interval.

Probability Mass Function (PMF)

The probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. It provides a mapping from each possible value of the random variable to its corresponding probability, ensuring that the total probability over all outcomes sums to 1.

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