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Lecture 16. Description of random variables: pdf, cdf. Expectation. Variance. Review: Area under a curve. Approximate by rectangles. Width  0, areas of rectangles  integral. Riemann integral, by definition, is the limit of such approximations. Mass of rod with uneven density.

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lecture 16

Lecture 16

Description of random variables: pdf, cdf.

Expectation. Variance.

review area under a curve
Review: Area under a curve

Approximate by rectangles.

Width  0,

areas of rectangles  integral.

Riemann integral, by definition, is the limit of such approximations

mass of rod with uneven density
Mass of rod with uneven density
  • Density (x), is a function of x.
  • The mass between x1 and x2, is the definite integral of (x) from x1 to x2.
  • If the difference (x2 - x1) is so small such that the density is roughly constant, the mass between x1 and x2 is approximately (x1)(x2 - x1).

x

center of gravity

M is the total mass.

Center of Gravity

string

  • Center of mass, or center of gravity, is the point where we can hold the rod in equilibrium (an unstable equilibrium).
moment of inertia
Moment of Inertia
  • The second moment

also have mechanical meaning.

M is the total mass.

simulation of continuous by digital computer
Simulation of continuous by digital computer
  • is never exact.
  • provides good approximation if precision is high enough.
mass vs probability
Mass vs Probability
  • Probability density function can be viewed as the density of a rod with unit mass.
  • Mass of a single point with zero length is zero.
  • The “center of gravity” of a pdf is the expectation of the random variable.
  • The “moment of inertia” is the second moment.
cumulative density function of discrete random variable
Cumulative density function of discrete random variable
  • F(x) =def P(Xx).

Example: Bernoulli random variable

P(X=0) = p, P(X=1)=1-p.

F(x)

1

p

x

1

derivative

f(x) = p(x)+(1-p) (x-1)

0

1

Derivative?
  • Derivative of discontinuous function does not exist.
  • However, if we allow generalized functions, such as the unit impulse function, also known as Dirac delta function, than we can talk about the pdf of discrete random variables.
  • The pdf of Bernoulli random variable is the sum of two delta functions.
cumulative density function of continuous random variable
Cumulative density function of continuous random variable
  • Cdf F(x) of continuous random variable are differentiable (or piecewise differentiable).
  • The derivative f(x) of cdf is the pdf.
  • Fundamental theorem of calculus:

(It is also equal to P(a<Xb) as P(X=a)=0.)

example uniform r v
Example: Uniform r.v.

F(x)

1

  • Cdf and pdf carry the same information.
  • Usually we use pdf in computation.
  • Cdf are found in tables of probability distribution.

1

x

f(x)

1

x

1

how to generate random variable
How to generate random variable?
  • If the cdf has a simple form, then we can use the inverse transform method.
  • To generate a random variable with cdf F(x)
    • Compute the inverse of F.
    • Generate a uniform random variable U between 0 and 1.
    • Return F-1(U).
expectation
Expectation
  • Suppose the pdf of a random variable X is fX(x). The expectation of X is defined as

The integral reduces to summation if X is a discrete

random variable, i.e., when fX(x) is a sum of delta functions.

easy properties
Easy properties
  • E[X+b]=E[X]+b.
  • E[aX] = aE[X].
  • E[g(X)]g(E[X]) in general.
  • If g(x)h(x) for all x, than E[g(X)] E[h(X)].
second moment and variance
Second moment and variance
  • Second moment of a random variable X is the expectation of X2.
    • E[X2]
  • Variance measures the level of variation from the mean. It is defined as the expectation of the square of deviation from the mean,
    • Var(X)=E[(X-E[X])2].
properties
Properties
  • Var(X) = E[X2]-(E[X])2.
  • Var(X+b) = Var(X).
  • Var(aX) = a2 Var(X).
what is the pdf of a function of x
What is the pdf of a function of X?
  • Let fX(x) be the pdf of random variable X.
  • What is the pdf of
    • X+b?
    • aX?
    • X2?
    • g(X), where g(x) is a monotonically increasing function?