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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> 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 ite...") |
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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 | |||
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>. | <math>p(d) = P(X = d)</math>. | ||
<ul><li> Show that | <ul style="list-style-type:lower-alpha"><li> Show that | ||
<math display="block"> | <math display="block"> | ||
Latest revision as of 01:19, 14 June 2024
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]]