The following table gives the probability distribution...

$$ \begin{aligned} &\text { The following table gives the probability distribution of a discrete random variable } x \text { . }\\ &\begin{array}{l|llllll} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & .03 & .17 & .22 & 31 & .15 & .12 \\ \hline \end{array} \end{aligned} $$ Find the following probabilities. a. $P(x=1)$ b. $P(x \leq 1)$ c. $P(x \geq 3)$ d. $P(0 \leq x \leq 2)$ e. Probability that $x$ assumes a value less than 3 f. Probability that $x$ assumes a value greater than 3 g. Probability that $x$ assumes a value in the interval 2 to 4

$$
\begin{aligned}
&\text { The following table gives the probability distribution of a discrete random variable } x \text { . }\\
&\begin{array}{l|llllll}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline P(x) & .03 & .17 & .22 & 31 & .15 & .12 \\
\hline
\end{array}
\end{aligned}
$$
Find the following probabilities.
a. $P(x=1)$
b. $P(x \leq 1)$
c. $P(x \geq 3)$
d. $P(0 \leq x \leq 2)$
e. Probability that $x$ assumes a value less than 3
f. Probability that $x$ assumes a value greater than 3
g. Probability that $x$ assumes a value in the interval 2 to 4

Key Concepts

Discrete Random Variable

A discrete random variable is one that can assume only a countable number of distinct values, each of which is associated with a probability. This concept is fundamental in probability theory and statistics where outcomes of processes or experiments are not continuous but rather take on specific, individual values.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of the discrete random variable. It provides a complete description of the likelihood of each outcome and is characterized by the fact that the sum of all probabilities for possible outcomes equals one.

Probability Mass Function (PMF)

The PMF is a function that gives the probability that a discrete random variable is exactly equal to a particular value. It is essential for calculating the probability of specific outcomes and serves as the foundation for further probability calculations involving the variable.

Cumulative Probability

Cumulative probability refers to the sum of the probabilities of all outcomes up to and including a certain value. This concept is used to compute the likelihood of the random variable falling within a specific range or meeting certain inequality criteria.

Interval Probability Calculation

Interval probability calculation involves finding the probability that a random variable lies within a given interval. For discrete variables, this is done by summing the individual probabilities (from the PMF) of all outcomes that fall within the specified interval, thus providing a measure of the likelihood for that range.

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