The random variable X has the probability distribution given...

The random variable $X$ has the probability distribution given by the rule $$ P(X=a)= \begin{cases}1 / k & \text { for } a=1,2,3,4,5, \\ 0 & \text { elsewhere } .\end{cases} $$ (a) is $X$ discrete or continuous? (b) find $k$ (c) graph the distribution of $X$ in two ways (d) find the distribution of $Y=(X-3)^2$ and graph it.

The random variable $X$ has the probability distribution given by the rule
$$
P(X=a)= \begin{cases}1 / k & \text { for } a=1,2,3,4,5, \\ 0 & \text { elsewhere } .\end{cases}
$$
(a) is $X$ discrete or continuous?
(b) find $k$
(c) graph the distribution of $X$ in two ways
(d) find the distribution of $Y=(X-3)^2$ and graph it.

Key Concepts

Discrete Random Variable

A discrete random variable is one that can take on a countable number of distinct outcomes. This concept distinguishes it from continuous random variables, which can take on any value within an interval. Understanding the nature of a random variable as discrete is crucial when applying summation rules for probabilities and when interpreting probability mass functions.

Normalization of Probability Distributions

Normalization is the process of ensuring that the total probability over all possible outcomes of a random variable sums to one. This usually involves calculating a constant, often denoted as k, such that when each individual probability is multiplied by 1/k (or another appropriate factor), their sum equals one. This principle is fundamental to defining valid probability distributions.

Graphical Representation of Discrete Distributions

Graphing a discrete probability distribution typically involves plotting each possible outcome along the x-axis and representing its probability on the y-axis, often using bar graphs or stem plots. This visual representation assists in understanding and communicating the distribution's characteristics, such as its concentration of probability mass and variability.

Transformation of Random Variables

The transformation of random variables involves creating a new random variable by applying a function to an existing one. Determining the distribution of the transformed variable requires mapping the probabilities from the original variable to the new one and, if necessary, adjusting for any many-to-one relationships. This process is essential for analyzing how modifications to data affect its distribution.

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