BBy Bot
Jun 09'24

Exercise

A manufactured lot of brass turnbuckles has [math]S[/math] items of which [math]D[/math] are defective. A sample of [math]s[/math] items is drawn without replacement. Let [math]X[/math] be a random variable that gives the number of defective items in the sample. Let [math]p(d) = P(X = d)[/math].

  • Show that
    [[math]] p(d) = \frac{{D \choose d} {{S - D} \choose {s - d}}}{{S \choose s}}\ . [[/math]]
    Thus, X is hypergeometric.
  • Prove the following identity, known as Euler's formula:
    [[math]] \sum_{d = 0}^{\min(D,s)}{ D \choose d} {{S - D} \choose {s - d}} = {S \choose s}\ . [[/math]]