Randomness in Computing L ECTURE 8 Last time Variance, covariance - - PowerPoint PPT Presentation

2/19/2020

Randomness in Computing

LECTURE 8

Last time

• Variance, covariance
• Chebyshev’s inequality
• Variance of Binomial and

Geometric RVs

In Discussion

• Median of a RV

Today

• Computing the median of an

array

Sofya Raskhodnikova;Randomness in Computing

Median of a random variable

• A value 𝑛 is the median of a random variable 𝑌

if Pr 𝑌 ≤ 𝑛 ≤ 1/2 and Pr 𝑌 ≥ 𝑛 ≤ 1/2.

• Example 1: 𝑌 is uniform over 𝑦1, … , 𝑦2𝑙+1,

where 𝑦1 < ⋯ < 𝑦2𝑙+1. What is the median?

• Example 2: 𝑌 is uniform over 𝑦1, … , 𝑦2𝑙,

where 𝑦1 < ⋯ < 𝑦2𝑙. Find all medians.

2/19/2020

Sofya Raskhodnikova; Randomness in Computing

Median and mean: another view

• Theorem. For a random variable X

with a finite expectation 𝜈 and a finite median 𝑛,

• 1. the expectation 𝜈 is the value of c

that minimizes the expression 𝐹 𝑌 − 𝑑 2 ;

• 2. the median 𝑛 is a value of 𝑑

that minimizes the expression 𝐹 |𝑌 − 𝑑| .

2/19/2020

Sofya Raskhodnikova; Randomness in Computing

Median and mean are close

• Theorem. For a random variable X with

expectation 𝜈, median 𝑛, and standard deviation 𝜏, 𝜈 − 𝑛 ≤ 𝜏.

2/19/2020

Sofya Raskhodnikova; Randomness in Computing

Randomized Algorithm for the Median of an Array

• On the board

2/19/2020

Sofya Raskhodnikova; Randomness in Computing