18.44 Lecture Notes - Lecture 17: Random Variable, Press Kit, Public Radio Exchange
Lecture 17: Uniform Random Variables
Recall Continuous Random Variable Definitions
!Say X is a continuous random variable if there exists a probability density function f = f on such that
!P{X∈B}= ∫ f(x)dx := ∫1 (x)f(x)dx#
!We may assume ∫ f(x)dx = ∫ f(x)dx = 1 and f is non-negative
!Probability of interval [a,b] is given by ∫ f(x)dx, the area under f between a and b
!Probability of any single point is zero
!Define cumulative distribution function: F(a) = F (a) := P{X <a} = P{X ≤ a} = ∫ f(x)dx#
Uniform measure on [0, 1]
!Suppose X is a random variable with probability density function f(x) =
!Then for any 0 ≤ a ≤ b ≤ 1 we have P{X [a, b]} = b - a
!Intuition: all locations along the interval [0, 1] equally likely
!Say that X is a uniform random variable on [0, 1] or that X is sampled uniformly from [0, 1]
Properties of Uniform Random Variable on [0, 1]
!Suppose X is a random variable with probability density function f(x) =
!What is E[X]?
!!Guess: 1/2 because 1/2 is in the middle of 0 and 1
!!Answer: ∫ f(x)dx = ∫ xdx = = 1/2
!What would you guess the variance is?
!!Expected square of distance from 1/2?
!!It is obviously less than 1/4, but how much less?
!E[X²] = ∫ f(x)x²dx = ∫ x²dx = = 1/3
!!So Var [X] = E[X²] - (Ε[Χ])² = 1/3 - 1/4 = 1/12
!Suppose X is a random variable with probability density function f(x) =
!What is the cumulative distribution function F (a) = P{X < a}
!!! 0 !a < 0
!!F (a) = a!a [0, 1]
!!! 1!a > 1
!What is the general moment E[X ] for k ≥ 0?
Uniform Random Variables on [α, β]
!Fix α < β and suppose X is a random variable with probability density function f(x) =
!!Then for any α < a < b < β we have P{X [a, b]} = (b - a)/(β - α)
!Intuition: all locations along the interval [α, β] are equally likely
!Say that X is a uniform random variable on [α, β] or that X is sampled uniformly from [α, β]
Properties of Uniform Random Variable on [0, 1]
!Suppose X is a random variable with probability density function f(x) =
!What is E[X]?
!!Intuitively, we'd guess the midpoint (α + β)/2
!!What is the cleanest way to prove this?
!!!One approach: Let Y be uniform on [0, 1] and try to show that X = (β - α)Y + α is uniform on [α, β]
!!!Then linearity of E[X] = (β - α)E[Y] + α = (1/2)(β - α) + α = (α + β)/2
!Using similar logic, what is the variance Var[X]?
!!Answer: Var[X] = Var[(β - α)Y + α] = Var[(β - α)Y] = (β - α)²Var[Y] = (β - α)²/12
Uniform Random Variables and Percentiles
!Toss n = 300 million Americans into a hat and pull one out uniformly at random.
!Is the height of the person you choose a uniform random variable?
!!Maybe in an approximate sense?#
!!No#
!Is the percentile of the person I choose uniformly random? In other words, let p be the fraction of people left in the hat whose
!heights are less than that of the person I choose. Is p, in some approximate sense, a uniform random variable on [0, 1]?#
!!The way I defined it, p is uniform from the set {0,1/(n−1),2/(n−1),...,(n−2)/(n−1),1}. When n is large, this is kind of
!!like a uniform random variable on [0, 1].#
Approximately Uniform Random Variables
!Intuition: which of the following should give approximately uniform random variables?#
!1. Toss n = 300 million Americans into a hat, pull one out uniformly at random, and consider that person’s height (in
! centimeters) modulo one.#
Document Summary
Say x is a continuous random variable if there exists a probability density function f = f on such that. We may assume f(x)dx = f(x)dx = 1 and f is non-negative. Probability of interval [a,b] is given by f(x)dx, the area under f between a and b. De ne cumulative distribution function: f(a) = f (a) := p{x
