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TERM PAPER

OFMTH:202

TOPIC:Disuss the various distributions ofcontinuous random variable.

Submitted To: Submitted By:Tejinder Paul Singh Summit Sakhre

E2803A10

B.Tech(IT)10803929


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ACKNOWLEDGEMENT

First and foremost I thank my teacher Mr.

Tejinder Paul Singh who has assigned me this

term paper to bring out my creative

capabilities.

I express my gratitude to my parents for being a

continuous source of encouragement for all theirfinancial aid.

I would like to acknowledge the assistance

provided to me by the library staff of LOVELY

PROFESSIONAL UNIVERSITY.

My heartfelt gratitude to my class-mates and forhelping me to complete my work in time

Summit Sakhre


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INDEXRandom Variable

Intuitive Description

Real-valued Random Variable

Distribution Functions Of Random Variables

Functions of Random Variables

Equality in Distribution


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Discreet Random Variables

Continuous Random Variables

Independent Random Variables

Remark on Random Variables

Random variable

In mathematics a random variable is a variable whose value is a function of the outcome of astatistical experiment that has outcomes of equal probability e.g. the number returned by rolling adie. Random variables are used in the study of probability. They were developed to assist in theanalysis of games of chance, stochastic events, and the results of scientific experiments bycapturing only the mathematical properties necessary to answer probabilistic questions. Furtherformalizations have firmly grounded the entity in the theoretical domains of mathematics bymaking use of measure theory.

The language and structure of random variables can be grasped at various levels of mathematicalfluency. Beyond an introductory level, set theory and calculus are fundamental. The concept of a

random variable is closely linked to the term "random variety": a random variety is a particularoutcome of a random variable.

There are two types of random variables: discrete and continuous. A discrete random variablemaps events to values of a countable set (e.g., the integers), with each value in the range havingprobability greater than or equal to zero. A continuous random variable maps events to values ofan uncountable set (e.g., the real numbers). Usually in a continuous random variable theprobability of any specific value is zero, although the probability of an infinite set of values


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(such as an interval of non-zero length) may be positive. However, sometimes a continuousrandom variable can be "mixed", having part of its probability spread out over an interval like atypical continuous variable, and part of it concentrated on particular values, like a discretevariable. This categorization into types is directly equivalent to the categorization of probabilitydistributions.

A random variable has an associated probability distribution and frequently also a probabilitydensity function. Probability density functions are commonly used for continuous variables.

Intuitive description

A random variable can be thought of as an unknown value that may change every time it isinspected. Thus, a random variable can be thought of as a function mapping the sample space ofa random process to the real numbers. A few examples will highlight this.

Examples

For a coin toss, the possible events are heads or tails. The number of heads appearing in one faircoin toss can be described using the following random variable:

and if the coin is equally likely to land on either side then it has a probability mass function

given by:

A random variable can also be used to describe the process of rolling a fair dice and the possibleoutcomes. The most obvious representation is to take the set {1, 2, 3, 4, 5, 6} as the samplespace, defining the random variable X as the number rolled. In this case ,


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An example of a continuous random variable would be one based on a spinner that can choose areal number from the interval [0, 2), with all values being "equally likely". In this case, X = thenumber spun. Any real number has probability zero of being selected. But a positive probabilitycan be assigned to any range of values. For example, the probability of choosing a number in [0,] is . Instead of speaking of a probability mass function, we say that the probability density ofX is 1/2. The probability of a subset of [0, 2) can be calculated by multiplying the measure ofthe set by 1/2. In general, the probability of a set for a given continuous random variable can becalculated by integrating the density over the given set.

An example of a random variable of mixed type would be based on an experiment where a coinis flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X

= 1; otherwise X = the value of the spinner as in the preceding example. There is a probabilityof that this random variable will have the value 1. Other ranges of values would have half theprobability of the last example.

Real-valued random variables

Typically, the observation space is the real numbers with a suitable measure. Recall,is the probability space. For real observation space, the function is a real-valued

random variable if

This definition is a special case of the above because generates the Borelsigma-algebra on the real numbers, and it is enough to check measurability on a generating set.


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.)

Distribution functions of random variables

Associating a cumulative distribution function (CDF) with a random variable is a generalizationof assigning a value to a variable. If the CDF is a (right continuous) Heaviside step function thenthe variable takes on the value at the jump with probability 1. In general, the CDF specifies theprobability that the variable takes on particular values.

If a random variable defined on the probability space is given, we canask questions like "How likely is it that the value of X is bigger than 2?". This is the same as the

probability of the event which is often written as for short, and

easily obtained since

Recording all these probabilities of output ranges of a real-valued random variable X yields theprobability distribution of X. The probability distribution "forgets" about the particularprobability space used to define X and only records the probabilities of various values of X. Sucha probability distribution can always be captured by its cumulative distribution function

and sometimes also using a probability density function. In measure-theoretic terms, we use therandom variable X to "push-forward" the measure P on to a measure dF on R. The underlyingprobability space is a technical device used to guarantee the existence of random variables,

and sometimes to construct them. In practice, one often disposes of the space altogether andjust puts a measure on R that assigns measure 1 to the whole real line, i.e., one works withprobability distributions instead of random variables.

Functions of random variables

If we have a random variable on and a Borel measurable function , then

will also be a random variable on , since the composition of measurable functions

is also measurable. (However, this is not true if f is Lebesgue measurable.) The same procedurethat allowed one to go from a probability space to can be used to obtain thedistribution of . The cumulative distribution function of is

If function f is invertible, i.e. f 1 exists, then the previous relation can be extended to obtain


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and, again with the same hypotheses of inevitability of f, we can find the relation between theprobability density functions by differentiating both sides with respect to y, in order to obtain

.

If there is no invertibility but each y admits at most a countable number of roots (i.e. a finitenumber of xi such that y = f(xi)) then the previous relation between the probability densityfunctions can be generalized with

where xi = fi1(y).

Example 1

Let X be a real-valued, continuous random variable and let Y = X2.

If y < 0, then P(X2 y) = 0, so

If y 0, then

so

Example 2

Suppose is a random variable with a cumulative distribution


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where is a fixed parameter. Consider the random variable Then,

The last expression can be calculated in terms of the cumulative distribution of X, so

Equality in distribution

Two random variables X and Y are equal in distribution if they have the same distributionfunctions:

Two random variables having equal moment generating functions have the same distribution.

This provides, for example, a useful method of checking equality of certain functions of i.i.d.random variables.

which is the basis of the KolmogorovSmirnov test.

Equality in mean

Two random variables X and Y are equal in p-th mean if the pth moment of |X Y| is zero, that

is,

As in the previous case, there is a related distance between the random variables, namely


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This is equivalent to the following:

Almost sure equality

Two random variables X and Y are equal almost surely if, and only if, the probability that they

are different is zero:

For all practical purposes in probability theory, this notion of equivalence is as strong as actualequality. It is associated to the following distance:

where 'sup' in this case represents the essential supremum in the sense of measure theory.

Equality

Finally, the two random variables X and Y are equal if they are equal as functions on theirprobability space, that is,

Discrete Random Variable

A discrete random variable is one which may take on only a countable number of distinct values

such as 0, 1, 2, 3, 4, ... Discrete random variables are usually (but not necessarily) counts. If a

random variable can take only a finite number of distinct values, then it must be discrete.

Examples of discrete random variables include the number of children in a family, the Friday

night attendance at a cinema, the number of patients in a doctor's surgery, the number of

defective light bulbs in a box of ten.

Continuous Random Variable

A continuous random variable is one which takes an infinite number of possible values.Continuous random variables are usually measurements. Examples include height, weight, theamount of sugar in an orange, the time required to run a mile.

Independent Random Variables


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Two random variables X and Y say, are said to be independent if and only if the value of X hasno influence on the value of Y and vice versa.

The cumulative distribution functions of two independent random variables X and Y are relatedby

F(x,y) = G(x).H(y)whereG(x) and H(y) are the marginal distribution functions of X and Y for all pairs (x,y).

Knowledge of the value of X does not effect the probability distribution of Y and vice versa.Thus there is no relationship between the values of independent random variables.

For continuous independent random variables, their probability density functions are related byf(x,y) = g(x).h(y)

whereg(x) and h(y) are the marginal density functions of the random variables X and Y

respectively, for all pairs (x,y).For discrete independent random variables, their probabilities are related byP(X = xi ; Y = yj) = P(X = xi).P(Y=yj)

for each pair (xi,yj).

Sums and Products of Random Variables, Notation

Suppose X and Y are random variables on the same sample space S. Then X +Y , kX and XY arefunctionson S defined as follows (where s S):

(X + Y )(s) = X(s) + Y (s), (kX)(s) = kX(s), (XY )(s) = X(s)Y(s)More generally, for any polynomial or exponential function h(x, y, . . . , z), we define h(X, Y, . . ., Z) to be thefunction on S defined by[h(X, Y, . . . , Z)](s) = h[X(s), Y(s), . . . , Z(s)]It can be shown that these are also random variables. (This is trivial in the case that every subsetof S is an event.)The short notation P(X = a) and P(a X b) will be used, respectively, for the probability thatX mapsinto a and X maps into the interval [a, b]. That is, for s S:P(X = a) P({s | X(s) = a}) and P(a X b) P({s | a X(s) b})

Analogous meanings are given to P(X a), P(X = a, Y = b), P(a X b, c Y d), and so on.

Probability Distribution of a Random Variable

Let X be a random variable on a finite sample space S with range space Rx= {x1, x2, . . . , xt}. Then Xinduces a function f which assigns probabilities pk to the points xk in Rx as follows:


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f (xk) = pk= P(X = xk) = sum of probabilities of points in S whose image is xk.

The set of ordered pairs (x1, f (x1)), (x2, f (x2)), . . . , (xt, f (xt )) is called the distribution of therandom variable

X; it is usually given by a table as i. This function f has the following two properties:(i) f (xk) 0 and (ii)_kf (xk) = 1Thus RX with the above assignments of probabilities is a probability space. (Sometimes we willuse the pairnotation [xk, pk] to denote the distribution of X instead of the functional notation [x, f (x)]).Distribution f of a random variable X

Theorem :Let S be an equiprobable space, and let f be the distribution of a random variable Xon S withthe range space RX= {x1, x2, . . . , xt}. Thenpi= f (xi ) = number of points in S whose image is xinumber of points in S

Remarks on Random Variables

1. You can think of a random variable as being analogous to a histogram. In a histogram,

you might show the percentage of your data that falls into each of several categories.

For example, suppose you had data on family income. You might find that 20 percent offamilies have an income below $30K, 27 percent have an income between $30 and $40k,21 percent have an income between $40 and $50k, and 32 percent have an income over$50k. A histogram would be a chart of that data with the income ranges on the X-axis andthe percentages on the Y-axis.

Similarly, a graph of a random variable shows the range of values of the random variableon the X-axis and the probabilities on the Y-axis. Just as the percentages in a histogramhave to add to 100 percent, the probabilities in a graph of a random variable have to add

to 1. (We say that the area under the curve of a probability density function has to equalone). Just as the percentages in a histogram have to be non-negative, the probabilities ofthe values of a random variable have to be non-negative.

2. A probability distribution function (pdf) for a random variable X is an equation or set ofequations that allows you to calculate probability based on the value of x. Think of a pdfas a formula for producing a histogram. For example, if X can take on the values 1, 2, 3,4, 5 and the probabilities are equally likely, then we can write the pdf of X as:


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f(X) = .2 for X = 1, 2, 3, 4, or 5

The point to remember about a pdf is that the probabilities have to be nonnegative andsum to one. For a discrete distribution, it is straightforward to add all of the probabilities.For a continuous distribution, you have to take the "area under the curve." In practice,

unless you know calculus, the only areas you can find are when the pdf is linear. See theUniform Distribution.

Mean and Variance

The mean of a distribution is a measure of the average. Suppose that we had a spinner where thecircle was broken into three unequal sections. The largest section is worth 5 points, one smallsection is worth 2 points, and the remaining small section is worth 12 points. The spinner has aprobability of .6 of landing on 5, a probability of .2 of landing on 2, and a probability of .2 oflanding on 12. If you were to spin the spinner a hundred times, what do you think your averagescore would be? To calculate the answer, you take the weighted average of the three numbers,

where the weights are equal to the probabilities. See the table below.

X P(X) P(X)*X

3 .2 0.6

5 .6 3.0

12 .2 2.4

mX = X = E(X) 6.0

The Greek letter m is pronounced "meiuw" and mX is pronounced "meiuw of X" or "meiuw subX."X is pronounced "X bar."E(X) is pronounced "The expected value of X."mX, X, and E(X) are three ways of saying the same thing. It is the average value of X.

In our example, E(X) = 6.0, even though you could never get a 6 on the spinner. Again, youshould think of the mean or average as the number you would get if you averaged a large numberof spins. On your first spin, you might get a 12, and the average would start out at 12. As you

take more spins, you get other numbers, and the average gradually tends toward 6. Inmathematical terms, you could say that the average asymptotically approaches 6.

Another property of the distribution of a random variable is its average disperson around themean. For example, if you spin a 12, then this is 6 points away from the mean. If you spin a 2,then this is 4 points away from the mean. You could spin the spinner 100 times and calculate theaverage disperson around the mean. Mathematically, we calculate the average dispersion by


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taking the square of the differences from the mean and weighting the squared differences by theprobabilities. This is shown in the table below.

The Greek letter s is pronounced "sigma." It is a lower

case sigma. The expression "var(X)" is pronounced"variance of X."

Suppose that the values of X were raised to 4, 6, and 13.What do you think would happen to the mean of X? What do you think would happen to thevariance of X? Verify your guesses by setting up the table and doing the calculation.

See if you can come up with values of X that would raise the mean and raise the variance. See ifyou can come up with values of X that would raise the mean but lower the variance. Finally,suppose we leave the values of X the same. Can you come up with different values of P(X) thatkeep the same mean but lower the variance? Can you come up with values of P(X) that keep the

same mean but raise the variance?

Next, consider a weighted average of random variables. In terms of a histogram, suppose thatyou had two zip codes with different average incomes. If you wanted to take the overall meanincome of the population in both zip codes, you would have to weight the means by the differentpopulations in the zip codes.

For example, suppose that there are 6000 families in one zip code, with a mean income of $50k.Suppose that there are 4000 families in another zip code, with a mean income of $40k. Theoverall mean income is equal to (1/10,000)(6000 * $50 + 4000 *$40) = $46k.

Similarly, suppose that you take a weighted average of two random variables. Let W = aX + bY.Then the mean of W is equal to a times the mean of X plus b times the mean of Y.

References

http://en.wikipedia.org/wiki/Random_variable

http://www.stats.gla.ac.uk/steps/glossary/probab

ility_distributions.html

X P(X) X-X (X-X)2 P(X)*(X-X)2

3 .2 -3 9 1.8

5 .6 -1 1 0.612 .2 6 36 7.2

s2 = var(X) = E[(X-X)2] 9.6
http://en.wikipedia.org/wiki/Random_variablehttp://www.stats.gla.ac.uk/steps/glossary/probability_distributions.htmlhttp://www.stats.gla.ac.uk/steps/glossary/probability_distributions.htmlhttp://en.wikipedia.org/wiki/Random_variablehttp://www.stats.gla.ac.uk/steps/glossary/probability_distributions.htmlhttp://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html


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Schaums Outline of Discreet Mathematics

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