exercise:2b14b350c6: Difference between revisions
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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> For a sequence of Bernoulli trials, let <math>X_1</math> be the number of trials until the first success. For <math>j \geq 2</math>, let <math>X_j</math> be the number of trials after the <math>(j - 1)</math>st success until the <math>j</math>th...") |
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For a sequence of Bernoulli trials, let <math>X_1</math> be the number of trials until the first success. For <math>j \geq 2</math>, let <math>X_j</math> be the number of trials after the <math>(j - 1)</math>st success until the <math>j</math>th success. It can be shown that | |||
<math>X_1</math>, <math>X_2</math>, ... is an independent trials process. | |||
<ul style="list-style-type:lower-alpha"><li> What is the common distribution, expected value, and variance for <math>X_j</math>? | |||
of trials until the first success. For <math>j \geq 2</math>, let <math>X_j</math> be the number of trials | |||
after the <math>(j - 1)</math>st success until the <math>j</math>th success. It can be shown that | |||
<math>X_1</math>, <math>X_2</math>, | |||
<ul><li> What is the common distribution, expected value, and variance for <math>X_j</math>? | |||
</li> | </li> | ||
<li> Let <math>T_n = X_1 + X_2 +\cdots+ X_n</math>. Then <math>T_n</math> is the time until the | <li> Let <math>T_n = X_1 + X_2 +\cdots+ X_n</math>. Then <math>T_n</math> is the time until the | ||
Latest revision as of 22:21, 14 June 2024
For a sequence of Bernoulli trials, let [math]X_1[/math] be the number of trials until the first success. For [math]j \geq 2[/math], let [math]X_j[/math] be the number of trials after the [math](j - 1)[/math]st success until the [math]j[/math]th success. It can be shown that [math]X_1[/math], [math]X_2[/math], ... is an independent trials process.
- What is the common distribution, expected value, and variance for [math]X_j[/math]?
- Let [math]T_n = X_1 + X_2 +\cdots+ X_n[/math]. Then [math]T_n[/math] is the time until the [math]n[/math]th success. Find [math]E(T_n)[/math] and [math]V(T_n)[/math].
- Use the results of (b) to find the expected value and variance for the number of tosses of a coin until the [math]n[/math]th occurrence of a head.