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Random Variables. an important concept in probability. A random variable , X, is a numerical quantity whose value is determined be a random experiment. Examples Two dice are rolled and X is the sum of the two upward faces.

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Random Variables

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Random variables

Random Variables

an important concept in probability


Random variables an important concept in probability

A random variable , X, is a numerical quantity whose value is determined be a random experiment

Examples

  • Two dice are rolled and X is the sum of the two upward faces.

  • A coin is tossed n = 3 times and X is the number of times that a head occurs.

  • We count the number of earthquakes, X, that occur in the San Francisco region from 2000 A. D, to 2050A. D.

  • Today the TSX composite index is 11,050.00, X is the value of the index in thirty days


Random variables an important concept in probability

Examples – R.V.’s - continued

  • A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner.

point

X

  • A chord is selected at random from a circle. X is the length of the chord.

chord

X


Definition the probability function p x of a random variable x

Definition – The probability function, p(x), of a random variable, X.

For any random variable, X, and any real number, x, we define

where {X = x} = the set of all outcomes (event) with X = x.


Definition the cumulative distribution function f x of a random variable x

Definition – The cumulative distribution function, F(x), of a random variable, X.

For any random variable, X, and any real number, x, we define

where {X≤x} = the set of all outcomes (event) with X ≤x.


Random variables an important concept in probability

Examples

  • Two dice are rolled and X is the sum of the two upward faces. S , sample space is shown below with the value of X for each outcome


Graph

Graph

p(x)

x


The cumulative distribution function f x

The cumulative distribution function, F(x)

For any random variable, X, and any real number, x, we define

where {X≤x} = the set of all outcomes (event) with X ≤x.

Note {X≤x} =f if x < 2. Thus F(x) = 0.

{X≤x} ={(1,1)} if 2 ≤ x < 3. Thus F(x) = 1/36

{X≤x} ={(1,1) ,(1,2),(1,2)} if 3 ≤ x < 4. Thus F(x) = 3/36


Continuing we find

Continuing we find

F(x) is a step function


Random variables an important concept in probability

  • A coin is tossed n = 3 times and X is the number of times that a head occurs.

The sample Space S = {HHH (3), HHT (2), HTH (2), THH (2), HTT (1), THT (1), TTH (1), TTT (0)}

for each outcome X is shown in brackets


Graph probability function

Graphprobability function

p(x)

x


Graph cumulative distribution function

GraphCumulative distribution function

F(x)

x


Random variables an important concept in probability

Examples – R.V.’s - continued

  • A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner.

point

X

  • A chord is selected at random from a circle. X is the length of the chord.

chord

X


Random variables an important concept in probability

E

Examples – R.V.’s - continued

  • A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner.

point

X

S

An event, E, is any subset of the square, S.

P[E] = (area of E)/(Area of S) = area of E


Random variables an important concept in probability

S

The probability function

Thus p(x) = 0 for all values of x. The probability function for this example is not very informative


Random variables an important concept in probability

S

The Cumulative distribution function


Random variables an important concept in probability

S


The probability density function f x of a continuous random variable

The probability density function, f(x), of a continuous random variable

Suppose that X is a random variable.

Let f(x) denote a function define for -∞ < x < ∞ with the following properties:

  • f(x) ≥ 0

Then f(x) is called the probability density function of X.

The random, X, is called continuous.


Probability density function f x

Probability density function, f(x)


Cumulative distribution function f x

Cumulative distribution function, F(x)


Random variables an important concept in probability

Thus if X is a continuous random variable with probability density function, f(x) then the cumulative distribution function of X is given by:

Also because of the fundamental theorem of calculus.


Random variables an important concept in probability

Example

A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner.

point

X


Random variables an important concept in probability

Now


Random variables an important concept in probability

Also


Random variables an important concept in probability

Now

and


Random variables an important concept in probability

Finally


Graph of f x

Graph of f(x)


Discrete random variables

Discrete Random Variables


Random variables an important concept in probability

Recall

p(x) = P[X = x] = the probability function of X.

This can be defined for any random variable X.

For a continuous random variable

p(x) = 0 for all values of X.

Let SX ={x| p(x) > 0}. This set is countable (i. e. it can be put into a 1-1 correspondence with the integers}

SX ={x| p(x) > 0}= {x1, x2, x3, x4, …}

Thus let


Random variables an important concept in probability

Proof: (thatthe set SX ={x| p(x) > 0} is countable)

(i. e. can be put into a 1-1 correspondence with the integers}

SX = S1 S2 S3 S3  …

where

i. e.


Random variables an important concept in probability

Thus the elements of SX = S1 S2 S3 S3  …

can be arranged {x1, x2, x3, x4, … }

by choosing the first elements to be the elements of S1 ,

the next elements to be the elements of S2 ,

the next elements to be the elements of S3 ,

the next elements to be the elements of S4 ,

etc

This allows us to write


A discrete random variable

A Discrete Random Variable

A random variable X is called discrete if

That is all the probability is accounted for by values, x, such that p(x) > 0.


Discrete random variables1

Discrete Random Variables

For a discrete random variable X the probability distribution is described by the probability function p(x), which has the following properties


Graph discrete random variable

Graph: Discrete Random Variable

p(x)

b

a


Continuous random variables

Continuousrandom variables

For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following properties :

  • f(x) ≥ 0


Graph continuous random variable probability density function f x

Graph: Continuous Random Variableprobability density function, f(x)


Random variables an important concept in probability

A Probability distribution is similar to a distribution ofmass.

A Discrete distribution is similar to a pointdistribution ofmass.

Positive amounts of mass are put at discrete points.

p(x4)

p(x2)

p(x1)

p(x3)

x4

x1

x2

x3


Random variables an important concept in probability

A Continuous distribution is similar to a continuousdistribution ofmass.

The total mass of 1 is spread over a continuum. The mass assigned to any point is zero but has a non-zero density

f(x)


The distribution function f x

The distribution function F(x)

This is defined for any random variable, X.

F(x) = P[X ≤ x]

Properties

  • F(-∞) = 0 and F(∞) = 1.

Since {X ≤ - ∞} = f and {X ≤ ∞} = S

then F(- ∞) = 0 and F(∞) = 1.


Random variables an important concept in probability

  • F(x) is non-decreasing (i. e. if x1 < x2 then F(x1) ≤F(x2) )

If x1 < x2 then {X ≤ x2} = {X ≤ x1} {x1 < X ≤ x2}

Thus P[X ≤ x2] = P[X ≤ x1] + P[x1 < X ≤ x2]

or F(x2) = F(x1) + P[x1 < X ≤ x2]

Since P[x1 < X ≤ x2] ≥ 0 then F(x2) ≥F(x1).

  • F(b) – F(a) = P[a < X ≤ b].

If a < bthen using the argument above

F(b) = F(a) + P[a < X ≤ b]

Thus F(b) – F(a) = P[a < X ≤ b].


Random variables an important concept in probability

  • p(x) = P[X = x] =F(x) – F(x-)

Here

  • If p(x) = 0 for all x (i.e. X is continuous) then F(x) is continuous.

A function F is continuous if

One can show that

Thus p(x) = 0 implies that


Random variables an important concept in probability

For Discrete Random Variables

F(x) is a non-decreasing step function with

F(x)

p(x)


Random variables an important concept in probability

For Continuous Random Variables Variables

F(x) is a non-decreasing continuous function with

f(x) slope

F(x)

x


Some important discrete distributions

Some Important Discrete distributions


The bernoulli distribution

The Bernoulli distribution


Random variables an important concept in probability

  • Success (S)

  • Failure (F)

Suppose that we have a experiment that has two outcomes

These terms are used in reliability testing.

Suppose that p is the probability of success (S) and

q = 1 – p is the probability of failure (F)

This experiment is sometimes called a Bernoulli Trial

Let

Then


Random variables an important concept in probability

The probability distribution with probability function

is called the Bernoulli distribution

p

q = 1- p


The binomial distribution

The Binomial distribution


Random variables an important concept in probability

  • Success (S)

  • Failure (F)

Suppose that we have a experiment that has two outcomes (A Bernoulli trial)

Suppose that p is the probability of success (S) and

q = 1 – p is the probability of failure (F)

Now assume that the Bernoulli trial is repeated independently n times.

Let

Note: the possible valuesof X are {0, 1, 2, …, n}


Random variables an important concept in probability

For n = 5 the outcomes together with the values of X and the probabilities of each outcome are given in the table below:


Random variables an important concept in probability

For n = 5 the following table gives the different possible values of X, x, and p(x) = P[X = x]


Random variables an important concept in probability

For general n, the outcome of the sequence of n Bernoulli trails is a sequence of S’s and F’s of length n.

SSFSFFSFFF…FSSSFFSFSFFS

  • The value of X for such a sequence is k = the number of S’s in the sequence.

  • The probability of such a sequence is pkqn – k( a p for each S and a q for each F)

  • There are such sequences containing exactly k S’s

  • is the number of ways of selecting the k positions for the S’s. (the remaining n – k positions are for the F’s


Random variables an important concept in probability

Thus

These are the terms in the expansion of (p + q)n using the Binomial Theorem

For this reason the probability function

is called the probability function for the Binomial distribution


Random variables an important concept in probability

Summary

We observe a Bernoulli trial (S,F)n times.

Let X denote the number of successes in the n trials.

Then X has a binomial distribution, i. e.

where

  • p = the probability of success (S), and

  • q = 1 – p = the probability of failure (F)


Random variables an important concept in probability

Example

A coin is tossed n= 7 times.

Let X denote the number of heads (H) in the n = 7 trials.

Then X has a binomial distribution, with p = ½ and n = 7.

Thus


Random variables an important concept in probability

p(x)

x


Random variables an important concept in probability

Example

If a surgeon performs “eye surgery” the chance of “success” is 85%. Suppose that the surgery is perfomed n = 20 times

Let X denote the number of successful surgeries in the n = 20 trials.

Then X has a binomial distribution, with p = 0.85 and n = 20.

Thus


Random variables an important concept in probability

p(x)

x


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