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| Random Variable and Probability Distribution |
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| 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. |
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| Outcome : HH HT TH TT |
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| No. of heads (x) : 2 1 1 0 |
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| 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. |
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| Random variable (r.v) |
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| Let S be a sample space associated with a given random experiment. |
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| A real valued function X which assigns to
each wi Î S, a unique
real number. |
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| Note: |
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| There can be several r.v's associated with an experiment. |
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| A random variable which can assume only a finite number of values or countably infinite values is called a discrete random variable. |
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| 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. |
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| Continuous random variable |
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| A random variable which can assume all possible values between certain limits is called a continuous random variable. |
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| Discrete Probability Distribution |
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| A discrete random variable assumes each of its values with a certain probability, |
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| Let X be a discrete random variable which takes values x1, x2, x3,…xn where pi = P{X = xi} |
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| Then |
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| X : x1 x2 x3 .. xn |
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| P(X): p1 p2 p3 ..... pn |
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| is called the probability distribution of x. |
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| Note 1: |
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| In the probability distribution of x |
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| Note 2: |
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| P{X = x} is called probability mass function. |
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| Note 3: |
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| 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. |
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| Probability distribution of a continuous random variable |
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| Let X be continuous random variable which can assume values in the interval [a,b]. |
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| A function f(x) on [a,b] is called the probability density function if |
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| Let X be a discrete random variable which can assume values x1, x2, x3,…xn with probabilities p1, p2, p3 ….. pn respectively then |
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| (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 : |
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| Example: |
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| 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. |
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| Suggested answer: |
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| X is the number of aces drawn while drawing two cards from a pack of cards. |
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| 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. |
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| Therefore the r.v. X can take the values 0, 1, 2. |
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| = 0.1629 - 0.0236 |
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| = 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 |
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| (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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