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UNIT - I ONE DIMENSIONAL RANDOM VARIABLES E.MATHIVADHANA, M.Sc.,M.PHIL. ASSISTANT PROFESSOR DEPARTMENT OF MATHEMATICS IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

SYLLABUS: Random variables – Probability function – Moments – Moment generating functions and their properties – Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal Distributions – Functions of a Random Variable. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBABILITY BASIC DEFINITIONS PROBABILITY: It is the science of decision-making with calculated risks in the face of uncertainty. EXPERIMENT: It is a process which results in some well defined outcomes. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

RANDOM EXPERIMENT: Eg: Tossing a fair coin, Throwing a die ,Taking a card from a pack of cards are Random Experiment. SAMPLE SPACE: The set of all possible outcomes of a random experiment is called its Sample Space. It is denoted by S. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Eg: Random Experiment Sample Space S Tossing a coin once {H,T} Tossing a coin Twice {HH,TH,HT,TT} Throwing a die once {1,2,3,4,5,6} OUTCOME: The result of a random experiment is called an outcome. Eg: Head or tail TRIAL: Any particular performance of a random experiment is called a trial. Eg: Tossing a coin IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

EVENT: The outcomes of a trial are called events. Eg: Getting head or tail MATHEMATICAL PROBABILITY: The Probability of the occurrence of a event A is denoted by P[A] and defined as P[A] = n[A] / n[S] 0 ≤ P[A] ≤ 1 n[A] = number of favourable cases to A. n[S] = number of possible cases in S. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

INDEPENDENT EVENTS: If the Occurrence of any one of them does not depend on the occurrence of the other. Eg: Tossing a coin, the event of getting a head in the first toss is independent of getting a head in the second and subsequent throws. CONDITIONAL PROBABILITY: The Conditional Probability of an event B ,assuming that the event A has already happened , is denoted by P[B/A] and defined as IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

EXAMPLE FOR CONDITIONAL PROBABILITY: When a fair die is tossed ,what is the probability of getting 1 given that an odd number has been obtained. SOLUTION: Let S={1,2,3,4,5,6} ; n[S]=6 Let A be the probability of getting a odd number A={1,3,5} ; n[A]=3 P[A]=n[A] / n[S] =3/6 Let B be the probability of getting 1. B={1} IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBABILITY DISTRIBUTION FUNCTION OF X : If X is a random variable ,then the function F[x] defined as is called the distribution function of X. PROBABILITY DISTRIBUTION FUNCTION OF Y : If X is a random variable ,then the function F[y] defined as is called the distribution function of Y. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

RANDOM VARIABLES: If E is an experiment having sample space S, and X is a function that assigns a real number X(s) to every outcome s є S, then X(s) is called a random variable (r.v.) Random variables are two types: i) Discrete Random variable ii) Continuous Random variable Eg: Someone continuously shoot on the same target, until really shot and then stop. s is the number of shots，so s One shooting Two shooting.... n shooting...... X(s) 1 2 .... n ...... IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Random Variable • Let S be the sample space. • A random variable X is a function X: SReal Suppose we toss a coin twice. Let X be the random variable number of heads IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Random Variable(Number of Heads in two coin tosses) We also associate a probability with X attaining that value. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Random Variable(Number of Heads in two coin tosses) IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Discrete Random variable: If X is a random variable which can take a finite number or countably infinite number of values, X is called a Discrete Random variable. Eg: 1. The number shown when a die is thrown. 2. Number of transmitted bits received in error. Continuous Random variable: If X is a random variable which can take all values in an interval, then is called a Continuous Random variable. Eg: The length of time during which a vacuum tube installed in a circuit functions is a continuous RV. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Probability Mass Function (or) Probability Function: If X is a discrete RV with distinct values x1 x2 x3 ,… xn… then P ( X = xi) = pi, then the function p(x) is called the Probability Mass Function. Provided p(i = 1, 2, 3, …) satisfy the following conditions: 1. for all i, and 2. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBABILITY DENSITY FUNCTION FOR CONTINUOUS CASE: If X is a continuous r.v., then f(x) is defined the probability density function of X. Provided f(x) satisfies the following conditions, 1. 2. 3. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

CUMULATIVE DISTRIBUTION FUNCTION (OR) DISTRIBUTION FUNCTION OF D.R.V.: The cumulative distribution function F(x) of a discrete r.v. X with probability distribution p(x) is given by CUMULATIVE DISTRIBUTION FUNCTION OF C.R.V.: The cumulative distribution function of a continuous r.v. X is IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

EXPECTED VALUE OF X FOR D.R.V: The mean or expected value of the discrete random variable of X, denoted as μ or E(X) is The variance of X denoted as σ2 or V(X) is IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBLEMS UNDER DISCRETE RANDOM VARIABLES: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBLEM 1: [TYPE I] A r. v. X has the following probability functions Find (i) value of k (ii) P(1.5 < X < 4.5 / X > 2) (iii) if P(X ≤ a) > ½, find the minimum value of a. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Solution: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBLEM 2 A random Variable X has the following probability distribution. • Find • the value of k • Evaluate P[X<2] and P[-2<X<2] • (iii)Find the cumulative Distribution of X. • (iv)Evaluate the Mean and Variance of X. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

(iii) The Cumulative Distribution of X: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

(iv) Mean: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Variance: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBLEM 3 : [TYPE II] Obtain the probability function or probability distribution function from the following distribution IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Problem 3: • The diameter, say X of an electric cable, is assumed to be a continuous r.v. with p.d.f. : • f(x) = 6x(1-x), 0 ≤ x ≤ 1 • Check that the above is a p.d.f. • Compute P(X ≤ ½ | ⅓ ≤ X ≤ ⅔) • Determine the number k such that • P(X< k) = P(X > k) IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

rth MOMENT ABOUT ORIGIN The rth moment about origin of a r.v. X is defined as the Expected value of the rth power of the X IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

MOMENT GENERATING FUNCTION The moment generating function of a r.v. X (about origin) whose probability function f(x) is given by t is a real parameter and the integration or summation being extended to the entire range of x. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Moment Generating Function Problem: Let the random variable X has the p.d.f Find the Moment Generating Function, (i) mean, (ii) variance of X and also (iii) find P(x>1/2) IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

(ii) VARIANCE: IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

STANDARD DISTRIBUTIONS IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

DISCRETE DISTRIBUTION BINOMIAL DISTRIBUTION A random variable X is said to follow Binomial Distribution if it assumes only non- negative values and its probability mass function is given by P(X=x) =P(x) = ncxpxqn-x, x=0, 1, 2, 3…..n, q=1-p 0 , otherwise Where n and p are called parameter of the Binomial distribution. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

Mean of Binomial Distribution. Mean = E(X) = np Variance of Binomial Distribution. Variance = Var(X) = npq. Moment Generating Function (M.G.F) of a Binomial Distribution. M G F = Mx ( t ) = E ( etx ) = ( p et + q )n IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

PROBLEM Out of 800 families with 4 children each, how many families would be expected to have (i) two boys and 2 girls, (ii) atleast 1 boy (iii) atmost 2 girls and (iv) children of both sexes. Assume equal probabilities for boys and girls. IFETCE/H&S- II/MATHS/MATHIVADHANA/IYEAR/ M.E.(CSE)/I-SEM/MA7155/APPLIED PROBABILITY AND STATISTICS /UNIT–I/PPT/VER1.2

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