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Quantitative Techniques

chapter 7 random Variables

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- 1. Mr. Rohan Bhatkar
- 2. Content- Random Variables Probability Mass Function Discrete Random Variables Distribution Function Variance Expectation Continuous Random Variables
- 3. Random Variables Consider an experiment of throwing two dice. We know that this experiments has 36 outcomes . Let x be the sum of numbers on the uppermost faces. For ExampleThe value of x is 2 if outcome of the experiment is (1,1) & probability of this outcome is 1/36. In this case we can associate real number with each outcome of random experiment or group of outcomes of random experiment. Here X is called Random Variables.
- 4. Probability Mass Function If x is discrete random variables values x1, x2, x3, …., xn then probability of each value is described by a function called the Probability Mass Function. The probability that random variable X takes values xi is denoted by p(xi). IllustrationSuppose fair coin is marked 1 & 2, dice numbered 1, 2, 3, 4, 5 & 6 are thrown simultaneously then probability mass function of random variables X which is sum of numbers on coin & dice is obtained as under. The sample space is: S={(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)} Note that n(s) = 12 X: sum of numbers on coin & dice. X = 2, 3, 4, 5, 6, 7, 8 P[x=2] = {(1,1)} = 1/12 P[x=3] = {(1,2)(2,1)} = 2/12 P[x=4] = {(1,3)(2,2)} = 2/12 P[x=5] = {(1,4)(2,3)} = 2/12 P[x=6] = {(1,5)(2,4)} = 2/12 P[x=7] = {(1,6)(2,5)} = 2/12 P[x=8] = {(2,6)} = 1/12 Thus probability distribution of random variable X is as under: X=x 2 3 4 5 6 7 8 P[X=x] 1/12 2/12 2/12 2/12 2/12 2/12 1/12 Note that P[xi] ≥ 0 & ∑ (xi) = 1
- 5. Discrete Random Variable A random variable which can assume only a countable number of real values & the values taken by variables depends on outcome of random experiment is called Discrete Random Variables. For Example1. Number of misprints per page of book. 2. Number of heads in n tosses of fair coin. 3. Number of throws of dice to get first 6.
- 6. Distribution Function Let X be a random variable. The function F defined for all real values x by F(x) = P [X = x], For all real x is called Distribution Function. A distribution function is also called as Cumulative Probability Distribution Function. Properties OF Distribution Function: If F is the D.F. of random variable X & if a <b then P[a < X ≤ b] = F(b) – F(a). Values of all distribution functions lie between 0 & 1 i.e. 0 ≤ F(x) ≤ 1 for all x. All distribution functions are monotonically non-decreasing i.e. 0 < y then F(x) < F(y). F(- ∞) = lim F(x) = 0 x-∞ F(+ ∞) = lim F(x) = 1 x→∞ If X is discrete random variable then F(x) = ∑ P(xi) xi ≤ x If values of discrete random variable X are like x1 < x2 < x3 < x4 … then P(Xn+1) = F(Xn+1) – F(Xn) If x is discrete random variable then D.E. is step function.
- 7. IllustrationConsider probability distribution of random variable x X=x 1 2 3 4 5 6 P[X=x] 0.1 0.2 0.3 0.2 0.1 0.1 If F(x) is distribution function of random variable x then F(1) = P[x ≤ 1] = P(1) = 0.1 F(2) = P[x ≤ 1] = P(1) + P(2) = 0.1 + 0.2 = 0.3 F(3) = 0.1 + 0.2 + 0.3 = 0.6 F(4) = 0.1 + 0.2 + 0.3 + 0.2 = 0.8 F(5) = 0.1 + 0.2 + 0.3 + 0.2 + 0.1 = 0.9 F(6) = 1 Thus values of x & corresponding cumulative probability distribution function is as under: X=x 1 2 3 4 5 6 P[X=x] 0.1 0.3 0.6 0.8 0.9 1.0 We will have graphical representation of random variable X, & the Graph of D.E. 1.0 Probability Distribution 0.8 3 P(x) f(x) 2 Cumulative Probability Distribution 0.6 0.4 1 0.2 1 2 3 4 5 6 1 2 3 X→ X→ 4 5 6
- 8. Variance If X is discrete random variable then variance of x is given by Var (X) = E[X – X (X)]2 Note that Var (X) = µ2 Var (X) = µ2` - µ1`2 Var (X) = E (X2) – [E(X)] 2 The positive square root of the variance is known as standard deviation S D (X) = + √Var (X) IllustrationFor a random variable X, E (X) = 10 and Var (X) = 5. Find Var (3X + 5), Var(X – 2), Var (4X). Also find E(5X – 4), E(4X + 3). SolutionVar (3X + 5) = Var (3X) = 9 Var (X) = 9 x 5 = 45 Var (x - 2) = Var (X) = 5 Var (4X) = 16 Var (X) = 16 x 5 = 80 Var (5X - 4) = 5 E(X) – 4 Var (5X - 4) = 5 x 10 – 4 = 46 Var (4X + 3) = 4 E(X) + 3 Var (4X + 3) = 4 x 10 + 3 = 43
- 9. Properties OF Variance: 1. Variance is independent of change of origin. It means that if X is random variable & ‘a’ is constant then variances of X and new variable X + a are same i.e. Var (X) + Var (X + a) consider Var (X + a) = E [X + a – (E (X) + a)]2 Var (X + a) = E [X + a – E (X) + a]2 Var (X + a) = E [X –E (X)]2 Var (X + a) = Var (X). 2. Variance of random variable depends upon change of scale. i.e. Var (a X) = a2 Var (X) consider Var (a X) = E [aX – E (aX)]2 = E [aX – a E (X)]2 = E a2 [X –E (X)]2 = a2 E [X –E (X)]2 = a2 Var (X). 3. From property 1 and 2 we can find Var (a X + b), Var (a X + b) = Var (a X) by property 1 2 Var (X) =a by property 2 4. Variance of constant is 0. put a=0 in a x + b then Var (a X + b) = Var (b) but Var (a x + b) = a2 × Var (X) Var (a x + b) = 0 × Var (X) = 0 Var (b) = 0
- 10. Expectation Mathematical Expectation of Discrete Random Variable: Once we have determined probability distribution function P(x) & distribution function of discrete random variable X, we want to compute the mean or variance of random variable X, the mean or expected value of X is nothing but weighted average of value X where corresponding probabilities are taken as weights, thus if X takes values x1, x2,.. With corresponding probabilities p(x1), p(x2),… then mathematical expectation of X denoted by E(X) is given by E(X) = ∑ xi P(xi). E(X) is also mean of random variable X. E(X) exists if series on right hand side is absolutely convergent. IllustrationThe p.m.f. of a random variable X is given below. Find E(X). Hence E(2X + 5) & E(X - 5). X=x 1 2 3 4 5 6 P[X=x] 0.1 0.15 0.2 0.3 0.15 0.1 SolutionExpected value of random variable is given by: E(X) = ∑xi P[X = xi] = 1(0.1) + 2(0.15) + 3(0.2) + 4(0.3) + 5(0.15) + 6(0.1) = 0.1 + 0.3 + 0.6 + 1.2 + 0.75 + 0.6 E(X) = 3.55 E(2X + 5) = 2 E(X) + 5 = 2 E(3.55) + 5 = 12.1 E(X - 5) = E(X) - 5 = (3.55) - 5 = -1.45
- 11. Properties OF Expectation: If X1 & X2 are two random variables then E(X1 + X2) = E(X1 ) + E(X2). This result can be generalized for X1, X2, …, Xn i.e. n random variables E(X1 + X2 + …. + Xn ) = E(X1) + E(X2) + …. + E(Xn) If X & Y are independent random variable, E(XY) = E(X) E(Y) If X is a random variable & a is constant then E(a X) = a E(X) and E(X + a) = E(X) + a If X is a random variable and a & b are constants E[a (X) + b] = a E(X) + b If X is a random variable and a & b are constants & g (X) a function X is random variable then E[a g (X) + b] = a E[g (X)] + b If X1, X2, X3 ….,Xn are any n random variable and if a1, a2, ….,an are any n constants then a1 X1 + a2 X2 + …. An Xn is called linear combination of n variables & expectation of linear combination is given by E(a1 X1 + a2 X2 + ….. + an Xn) = = = If X ≥ 0 then E(X) ≥ 0. If X and Y are two random variables & if X ≤ Y then E(X) ≥ E(Y). If X & Y are independent random variable then E[a g (X) + h (Y)] = E[g (X)] E[h (Y)] then g (X) is a function of X & its random variable. Also h (Y) is a function of Y & random variable.
- 12. Continuous Random Variables Consider the small interval (X, X + dx) of length dx round the point x. Let f(x) be any continuous function of x so that f(x) represent the probability that falls in very small interval (X, x + dx) Symbolically P[x ≤ X ≤ x + dx] = f(x) dx. y f(x) dx Y = f(x) x X – dx/2 X + dx/2 In the figure f(x) dx represents the area bounded by the curve y = f(x); X-axis a& the ordinates x and x + dx. The function f(x) so defined is known as probability density function of random variable X & usually abbreviated as p.d.f. The expression f(x) dx usually written as F (x), is known as probability differential & f(x) is known as probability density curve. The probability that X lie in the interval dx is f(x) dx. Thus the p.d.f. of random variable x is defined as f(x) = lim P [x ≤ X ≤ x + δx] δx → 0 δx
- 13. IllustrationA random variable X has following p.d.f. f(x) = K -∞<x<∞ 2 1+x =0 Otherwise Find k: Solution: Since X is continuous random variable with density function f(x), ∞ ⌠ f(x) dx =1 -∞ ∞ ⌠ -∞ k[tan-1 K dx 2 1+x ∞ x] -∞ =1 =1 k[tan-1 ∞ - tan-1 ∞ (- ∞)] = 1 k[π/2 + (π/2)] k k π =1 =1 = 1/ π

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