Lecture 16

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# Lecture 16 - PowerPoint PPT Presentation

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

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
• 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

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
• The second moment

also have mechanical meaning.

M is the total mass.

Simulation of continuous by digital computer
• is never exact.
• provides good approximation if precision is high enough.
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
• 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

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
• 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.

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?
• 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
• 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
• 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 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
• 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?
• 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?