Final answer:
A random variable can be discrete or continuous depending on whether its values are countable or measurable. The given random variables can be classified as discrete or continuous and an explanation is provided for each one.
Step-by-step explanation:
A random variable in probability theory is a variable that can take on different values as a result of a random experiment or process. It represents the outcomes of the experiment or process, and the probability distribution of the random variable describes how likely each outcome is.
A discrete random variable can only take on a countable number of distinct values. It typically represents outcomes that can be counted, like the number of heads in multiple coin flips or the number of questions answered correctly on a test.
A continuous random variable can take on any value within a certain range. It typically represents measurements or quantities that can be measured, like temperature or length.
Now, let's analyze each of the given random variables:
a) Flip a coin three times. Let X be the total number of heads:
X is a discrete random variable, as it represents the countable outcomes of getting 0, 1, 2, or 3 heads.
b) Randomly select a student who took a true/false test with 100 questions. Let X be the number of questions answered correctly:
X is a discrete random variable, as it represents the countable outcomes of getting 0 to 100 questions answered correctly.
c) Randomly select a mutual fund. Let X be the number of companies in the portfolio:
X is a discrete random variable, as it represents the countable outcomes of the number of companies in the portfolio.
d) Randomly select 50 community college students. Let X be the exact average age of the group:
X is a continuous random variable, as it can take on any value within a range of possible average ages.
e) Randomly select a newborn baby. Let X be the exact length in inches:
X is a continuous random variable, as it can take on any value within a range of possible lengths.