Random Variables Assignment help




homework assignment help is most useful online help portal for the students that providing all Online Random Variables assignment help Services. In Probability and statistics, a random variable or stochastic variable is a variable whose value results from a measurement on some random process. A random variable is a function, which maps events or outcomes (e.g., the possible results of rolling two dice: (1, 1), (1, 2), etc.) to real numbers (e.g., their sum). A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the potential values of a quantity whose already-existing value is uncertain (e.g., as a result of incomplete information or imprecise measurements). Intuitively, a random variable can be thought of as a quantity whose value is not fixed, but which can take on different values; a probability distribution is used to describe the probabilities of different values occurring. Realizations of a random variable are called random variates.

Random Variable and Probability Distribution

Learn about random variables and probability distribution in this page and get quality statistics help.

If is often very important to allocate a numerical value to an outcome of a random experiment. For example consider an experiment of tossing a coin twice and note the number of heads (x) obtained.

Outcome : HH HT TH TT

No. of heads (x) : 2 1 1 0

x is called a random variable, which can assume the values 0, 1 and 2. Thus random variable is a function that associates a real number to each element in the sample space.

Random variable

Let S be a sample space associated with a given random experiment.

A real valued function X which assigns to each wi Î S, a unique real number.

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There can be several r.v's associated with an experiment.

A random variable which can assume only a finite number of values or countably infinite values is called a discrete random variable.

e.g., Consider a random experiment of tossing three coins simultaneously. Let X denote the number of heads then X is a random variable which can take values 0, 1, 2, 3.

Continuous random variable

A random variable which can assume all possible values between certain limits is called a continuous random variable.

Discrete Probability Distribution

A discrete random variable assumes each of its values with a certain probability,

Let X be a discrete random variable which takes values x1, x2, x3,…xn where pi = P{X = xi}

Then

X : x1 x2 x3 .. xn

P(X): p1 p2 p3 ..... pn

is called the probability distribution of x.

Note:

1)

In the probability distribution of x

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2) P{X = x} is called probability mass function.

3) Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can have a formula in the form of a function represented by f(x) usually called the probability density function.

Probability distribution of a continuous random variable

Let X be continuous random variable which can assume values in the interval [a,b].

A function f(x) on [a,b] is called the probability density function if

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Discrete Random Variable

Let X be a discrete random variable which can assume values x1, x2, x3,…xn with probabilities p1, p2, p3 ….. pn respectively then

(a) Mean of X or expectation of X denoted by E(X) or m is given by

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(b) Variance of X denoted by s2 is given by

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Note:

random variable

Example

Problem:

Two cards are drawn successively without replacement, from a well shuffled deck of cards. Find the mean and standard deviation of the random variable X, where X is the number of aces.

Solution:

X is the number of aces drawn while drawing two cards from a pack of cards.

The total ways of drawing two cards 52C2. Out of 52 cards these are 4 aces. The numbers of ways of not drawing an Ace =48C2. The number of ways of drawing an ace is 4C1 x 48C1 and two aces 4C2.

Therefore the r.v. X can take the values 0, 1, 2.

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= 0.1629 - 0.0236

= 0.13925

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Let X be a continuous random variable which can assume values in (a, b) and f(x) be the probability density of x then

(a) Mean of X or Expectation of X is given by

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(b) Variance of x is given by

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Types of Random Variables:

  • Discrete Random variable
  • Continuous Random variable

Discrete Random Variable:

If a random variable takes only fixed values, it is called a discrete random variable.

Continuous Random Variable :

A Random Variable of X is said to be continuous if it can take all possible values between certain given limits.

Problems on Distribution of Two Random Variables:

Sample problem in  distribution of two random variables 1:

Find out the following variables are discrete or continuous. probable values for the random variable.

 (a) The number of fans that burn out in a room of 10 light fans in the next year.

 Discrete x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

 (b) The number of leaves on a randomly selected Oak tree

 Discrete x = {0, 1, 2, 3…}

(c) The length of time between calls to 911.

 Continuous x > 0

 A probability distribution can be in the type of table, graph or mathematical formula.

Example in distribution of two random variables 2:

     3 letters are written to the three different men and addresses on three envelopes are also written. Without seeing at the addresses, what is the probability that (i) all the letters go into right envelopes, (ii) none of the letters goes into right envelopes?

Solution:

Let A, B and C are represented by the envelopes and 1, 2 and 3 denote the corresponding letters.

Let X represented by the event of putting the letters go into right envelopes.

Y represented by the event of putting none of the letters go into right envelope.

            S = {c1, c2, c3, c4, c5, c6}, n(S) = 6

            X = {c1}, n(X) = 1 Y = {c4, c5}, n(Y) = 2

        ∴ P(X) =1/6 P(Y) =2/6 =1/3.

Example in distribution of two random variables 3:

There are two coins are tossed simultaneously, what is the probability of getting

  • exactly one head

The sample space is S = {HH, HT, TH, TT}, n(S) = 4

Solution:        

Let A is  the event of getting one head, B is the event of getting atleast one head and C is the event of getting at-most one head.

              ∴  A = {HT, TH}, n(A) = 2

  •   P(A) = n(A) / n(S) = 2/4 = 1/2


Example in distribution of two random variables 4:

The table shows the probability distribution for the random variable x, where x represents the amount of CDs a individual rents from a film store during a distinct visit.

 

X            p(x)

 0          0.07

 1          0.59

 2           0.21

 3          0.10

 4          0.03

 5          0.01

Determine whether the following is a valid probability distribution for the random variable x.

No, since `sum` P(x) = .97 is not equal to 1.

x = (0) (.07) + (1) (.59) + (2) (0.21) + (4) (.03) + (5) (0.01)

          = 59.73 CDs                

Normal Random Variable:-->

Random variable

The random variable is a variable whose value is a function of the outcome of a statistical experiment random variable are used in the study of probability.

Normal random variable

The normal random variable is called as chi-squared random

The squared of the standard normal random variable is the chi-squared random variable of degree 1. Therefore, the sum of squares of independent standard normal random variables is the chi-squared random variable of degree.

Normal Random Variable Squared Explanation:

Here will explain the normal random variable

For any real numbers µ and σ >0, the Gaussian probability distribution function with mean µ and variance σ2is defined by

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When µ=0 and σ =1, it is regularly called standard normal distribution.

A random variable x having distribution density ƒ is said to be a normally distributed random variable, denoted by X˜N (μ, σ2). It has expected value µ, and variance σ2

Chi Squared Random Variable

Xis a central chi-squared random variable with n degrees of freedom if

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For x>0, where I represents the gamma function. Parameters: n ε N

Syntax:  X˜ X2 (n)

Normal Random Variable Squared Example Problem:

One example used to explain the normal random variable

Example:

Find Expected value of [random variable squared]. That is A[x²] .Given that X is a normal random variable with mean µ = 0, variance s² find That is A[x²]
solution:
Starting with a definition of variance:

Variance(X) = A[x²] - A[x]A[x] = s²

The problem stated tat the mean of X is 0 then:

           s²= A[x²] - 0*0
              = A[x²]
So in this case the expected value of the square of a random variable X, X~(N,0, s^2) is the variance of X = s²


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