This module introduces the concept in probability distributions, such as probability mass function(pmf), cumulative distribution function(cdf) and probability density function(pdf).
The distribution
of a random variable
is simply a
probability measure which assigns probabilities to events on thereal line. The distribution
answers questions of the form:
What is the probability that
lies in some subset
of the real
line?
In practice we summarize
by its
Probability Mass Function - pmf (for
discrete variables only),
Probability Density Function -
pdf (mainly for continuous variables), or
Cumulative Distribution Function - cdf (for either
discrete or continuous variables).
Probability mass function (pmf)
Suppose the discrete random variable
can take a set of
real values
, then the
pmf is defined as:
where
. e.g. For a normal 6-sided die,
and
. For a pair of dice being thrown,
and the pmf is as shown in (a) of
.
Cumulative distribution function (cdf)
The
cdf can describe discrete, continuous or
mixed distributions of
and is
defined as:
For discrete
:
giving step-like cdfs as in the example of (b) of
.
Properties follow directly from the Axioms of Probability:
,
is non-decreasing as
increases
where there is no ambiguity we will often drop the subscript
and refer to the cdf as
.
Probability density function (pdf)
The
pdf of
is
defined as the derivative of the cdf:
The pdf can also be interpreted in derivative form as
:
For a discrete random variable with pmf given by
:
An example of the pdf of the 2-dice discrete random process isshown in (c) of
.
(Strictly the delta functions should extend vertically toinfinity, but we show them only reaching the values of their
areas,
.)
The pdf and cdf of a continuous distribution (in this case the
normal or
Gaussian distribution) are
shown in (d) and (e) of
.
The cdf is the integral of the pdf and
should always go from zero to unity for a valid probabilitydistribution.
Properties of pdfs:
As for the cdf, we will often drop the subscript
and refer simply to
when no confusion can arise.
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?