A random variable is a quantitative variable whose values are based on chance. By “chance”, we mean that you can’t know the outcome before it occurs.

A discrete random variable is a random variable whose possible values can be listed.

• Variables: \(x\),\(y\),\(z\)

• Random variables: \(X\), \(Y\), \(Z\)

• The event that the random variable \(X\) equals \(x\): \(\{X=x\}\)

• The probability that the random variable \(X\) equals \(x\): \(P(X=x)\)

A probability histogram is a histogram where the heights of the bars correspond to the probability of each value.

• For discrete random variables, each “bin” is one of the listed values.

(Assume for the sake of the example that no one has more than 4 siblings.)

The mean of a discrete random variable \(X\) is denoted \(\mu_X\).

\[\mu_X = x_1P(X=x_1) + x_2P(X=x_2) + \dots + x_nP(X=x_n).\]

If it’s clear which random variable we’re talking about, we can just write \(\mu\).

\[\mu = 0(0.200)+1(0.425)+2(0.275)+3(0.075)+4(0.025)=1.3\]
Interpretation: in a large number of independent observations of a random variable \(X\), the mean of those observations will approximately equal \(\mu\).

The mean of a random variable is also called the expected value or expectation.

Since measures of center are meant to identify the most common or most likely, you can think of this as the value we expect to see (most often).

The larger the number of observations, the closer their average tends to be to \(\mu\). This is known as the law of large numbers.

Suppose I took a random sample of 10 people and asked how many siblings they have. \[2,2,2,2,1,0,3,1,2,0\] In my random sample of 10, \(\bar{x}=2\), which is a reasonable estimate but not that close to the true mean \(\mu=1.3\).

• A random sample of 30 gave me a mean of \(\bar{x}=1.53\).
• A random sample of 100 gave me a mean of \(\bar{x}=1.47\).
• A random sample of 1000 gave me a mean of \(\bar{x}=1.307\).

\[ \sigma_X^2 = [x_1^2P(X=x_1) + x_2^2P(X=x_2) + \dots + x_n^2P(X=x_n)]-\mu_X^2\]

As before, the standard deviation is the square root of the variance: \[\sigma = \sqrt{\sigma^2}\]

Calculate the standard deviation of the Siblings variable.

In general, a table is the best way to keep track of a variance calculation:

• The variance is \(\sigma^2 = 2.6 - 1.3^2 = 0.9\)
• The standard deviation is \(\sigma = \sqrt{0.9} = 0.9539\)