MA 305 Random Variables By Prof Nutan Patel

MA 305 Random Variables By: Prof. Nutan Patel Asst. Professor in Mathematics IT-NU A-203 patelnutan. wordpress. com MA 305 Mathematics for ICE 1

Markov chain is a sequence of experiments with the following properties: 1. An experiment has a finite number of discrete outcomes, called states. 2. With each additional trial the experiment can move from its present state to any state or remain in the same state. 3. The probability of going from one state to another on the next trial depends only on the present state and not on past states. 4. The probability of moving from any one state to another in one step is represented in a transition matrix. • The transition matrix is square, since all possible states are used for rows and columns. • Each entry is between 0 and 1, • The entries in each row add to 1. 5. The state matrix times the transition matrix give the next-state matrix. MA 305 Mathematics for ICE 2

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Random Variable • Definition: A random variable is a rule that assigns a number to each outcomes of an experiment, whose value is subject to variations due to chance. • In the inspection of a manufactured product we may be interested only in the number of defectives; in the analysis of a road test we may be interested only in the average speed and the average fuel consumption; and in study of the performance of a rotary switch we may be interested only in its lubrication, the current, and the humidity. All these numbers are associated with situations involving an element of chance- in other world, they are values of random variables. • Random variable is denoted by a capital letter such as X or Y. MA 305 Mathematics for ICE 4

Example-1. If a coin is tossed thrice, there are eight possible outcomes. We can assign a number to each outcome by simply giving the number of heads that appear in each case. The values of X are assigned as follows. • P(X=1)=3/8, P(X=3)=1/8. • P(X≤ 2)=7/8. • P(0<X<3)=6/8=3/4. OUTCOME X HHH 3 HHT 2 HTH 2 HTT 1 THH 2 THT 1 TTH 1 TTT 0 MA 305 Mathematics for ICE 5

Types of Random Variable There are two types of random variables. 1. Discrete random variable and 2. Continuous random variable. 1. Discrete random variable: A random variable which can take only finite or isolated values in a given interval is called discrete random variable. for example, the number of heads in tossing coins, the number of auto passengers can take on only the values 1, 2, 3, and so on. Discrete random variables can be measured exactly. MA 305 Mathematics for ICE 6

2. Continues random variable A random variable which can take all possible values that are infinite in a given interval is called continuous random variable. For example, Measuring the height of a student selected at random, Finding the average life of an MRF tyre etc. Continues random variables cannot be measured exactly. MA 305 Mathematics for ICE 7

Probability Distribution • A probability distribution is a listing of the probabilities of all possible outcomes if the experiment is performed. • This means that a probability distribution shows how the total probability 1 is distributed among the different possible outcomes of the experiment. Definition: If a random variable has the values a 1, a 2, …, ak then a probability distribution P(X) is a rule that assigns a probability P(ai) to each value ai. 1. 0≤ ai ≤ 1 2. P(a 1)+ P(a 2)+…+ P(ak) = 1. MA 305 Mathematics for ICE 8

Example: A coin is tossed thrice. P(0)=1/8. P(1)=3/8. P(2)=3/8. P(3)=1/8. We can represent a probability distribution graphically using histogram. Form a category for each value of the random variable. The probability of each value of X determines the height of bar. The graph of this probability distribution is 0. 4 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 P(0) P(1) P(2) Category 1 P(3) MA 305 Mathematics for ICE 9

Types of Probability Distributions Probability Distribution are classified as: 1. Discrete probability distribution and 2. Continuous probability distribution. 1. Discrete probability distribution : the probability distributions of discrete random variables are discrete probability distributions. Most widely used probability distributions of discrete random variables are: Binominal Distribution, Poisson Distribution. MA 305 Mathematics for ICE 10

2. Continuous probability distribution The probability distribution of continuous random variables are continuous probability distributions. One of the very widely used continuous probability distribution is “Normal Distribution”. MA 305 Mathematics for ICE 11

Probability Function If for random variable X, the real valued function f(x) is such that P(X=x) = f(x) Then f(x) is called probability function. ØIf X is a discrete random variable then its probability function f(x) is discrete probability function. It is called probability mass function. ØIf X is a continuous random variable then its probability function f(x) is continuous probability function. It is called probability density function. MA 305 Mathematics for ICE 12

Expected value • MA 305 Mathematics for ICE 13

• Example: Find the expected value of X, where the values of X and their corresponding probabilities are given as x 0 1 2 3 P(x) 1/8=0. 125 3/8=0. 375 1/8=0. 125 • E(x)=0+1*0. 375+2*0. 375+3*0. 125=1. 5 MA 305 Mathematics for ICE 14

Ex: A tray electronic components contains nine good components and three defective components. If two components are selected at random, what is the expected number of defective components? Ans: P(0)=12/22. P(1)=9/22 P(2)=1/22 E(x)=0. 5 MA 305 Mathematics for ICE 15