Biostatistics for Animal Science

Chapter 3
Random Variables and their Distributions
A random variable is a rule or function that assigns numerical values to observations or
measurements. It is called a random variable because the number that is assigned to the
observation is a numerical event which varies randomly. It can take different values for
different observations or measurements of an experiment. A random variable takes a
numerical value with some probability.
Throughout this book, the symbol y will denote a variable and yi will denote a
particular value of an observation i. For a particular observation letter i will be replaced
with a natural number (y1, y2, etc). The symbol y0 will denote a particular value, for
example, y ≤ y0 will mean that the variable y has all values that are less than or equal to
some value y0.
Random variables can be discrete or continuous. A continuous variable can take on all
values in an interval of real numbers. For example, calf weight at the age of six months
might take any possible value in an interval from 160 to 260 kg, say the value of 180.0 kg
or 191.23456 kg; however, precision of scales or practical use determines the number of
decimal places to which the values will be reported. A discrete variable can take only
particular values (often integers) and not all values in some interval. For example, the
number of eggs laid in a month, litter size, etc.
The value of a variable y is a numerical event and thus it has some probability. A table,
graph or formula that shows that probability is called the probability distribution for the
random variable y. For the set of observations that is finite and countable, the probability
distribution corresponds to a frequency distribution. Often, in presenting the probability
distribution we use a mathematical function as a model of empirical frequency. Functions
that present a theoretical probability distribution of discrete variables are called probability
functions. Functions that present a theoretical probability distribution of continuous
variables are called probability density functions.
3.1 Expectations and Variances of Random Variables
Important parameters describing a random variable are the mean (expectation) and
variance. The term expectation is interchangeable with mean, because the expected value of
the typical member is the mean. The expectation of a variable y is denoted with:
E(y) = µy
The variance of y is:
Var(y) = σ 2 y = E[(y – µy) 2 ] = E(y 2 ) – µy 2
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