Binomial Distribution - lec. 10

ucla | MATH 170E | 2023-02-01T11:58

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Table of Contents

• Definitions
• Big Ideas
  • Binomial Distribution
• Resources

Definitions

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Big Ideas

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Binomial Distribution

Number of success in fixed number of trials

• A Binomial distribution is the distribution of $n\ge 1$ independent, identical Bernoulli trials that we write $X\sim \text{Binomial(n,p)}$ where $n$ is the number of trials and $p$ is the probability of each success (prob. of success of 1 trial)
• then the PMF is

$p_X(x)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}x\in 0,1,\dots ,n$

• CDF

$\text{Missing end{cases}}$

• then the MGF is

$M_X(t)=\mathbb{E}[e^{tX}]=(1-p+p{e}^t{)}^n\mathrm{\forall }t\in \text{R}$

• then the expected value (mean) is

$\mathbb{E}[X]=np$

• then the variance is

$\text{var}(X)=np(1-p)$

Resources

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**SUMMARY
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