Chapter 7 Statistics Class 11 Notes
Important Notes of complete Chapter 7 Statistics Class 11 Notes written by Professor Mr. Faraz Qasir Suib. These notes are very helpful in the preparation of Chapter 7 Random Variables 11 Notes PDF for the students of the intermediate and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Our comprehensive Chapter 7 Statistics Class 11 Notes will ensure you're fully prepared for your exams.
- What is a random variable?
- Define continuous and discrete random variable.
- What is distribution function of a discrete random variable X?
- Define probability density function.
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- What are the properties of discrete probability distribution?
- What is meant by mathematical expectation of a random variable?
- Enlist properties of expectations.
- Given X = 0, 1, 2 and P(X) = 9/16, 6/16, 1/16, find variance of X.
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- Given the probability distribution. Find K
- Given that f(x) = x/10, x = 1, 2, 3, 4. Show that f(x) is a probability function.
- Given E(X) = 0.63 and Var(X) = 0.2331 then find E(X2 ).
- Given X = 1, 2, 3, 4, 5 and P(X) = 1/10, 3/10, P, 2/10, 1/10. Find the value of P.
- Find the probability distribution of the number of heads when two coins are tossed.
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- Define random experiment.
- Enlist properties of probability mass function.
- If E(X) = 1.4, then find E(5x – 4).
- Given: E(x) = 0.56, var(x) = 1.36 and if y =2x + 1, then find E(y) and var(y).
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- Introduction to Random Variables: Every random experiment results in two or more outcomes and usually the interest is in a particular aspect of the outcomes of the experiment. For example, when a pair of dice is thrown, the interest may be in the total of upturned dots on both dice.
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- In case of this experiment, the total may be 2 (one on each die), 3 (one on the first die and two on the other) and so on. It may be 4, 5, 6, 7, 8, 9, 10, 11 or 12. In the language of probability, these values associated with outcomes are the values of a so-called random variable. An other example is the total number of children in each of the fifty randomly chosen families. If no family has more than 5 children then the values of this random variable i.e., number of children in each family, would be 0,1,2,3,4,5 i.e., no child, one child, two children, 3 children, 4 children and five children respectively.
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- A variable whose values depend upon the outcomes of a random experiment is called a random variable. We will denote the random variable by the capital letters X, Y or Z and their values by the corresponding small letters x, y or z.
- Example 7.1: Let a pair of dice be thrown and Y denote the random variable that is the sum of upturned values on the two dice. There are 36 outcomes and Y assigns to the outcome (1, 1) the real number 1+1=2. It assigns to the outcome (2, 1), the real number 2+1=3 and so on uptil the outcome (6, 6), the real number 6 + 6 = 12 so, the values assigned are 2,3,4,5,6,7,8,9,10,11 and 12.
- Random numbers and their generation: Random numbers are a sequence of digits from the set {0, 1, 2, ...., 9}. So that, at each position in the sequence, each digit has the same probability 0.1 of being selected irrespective of the actual sequence, so far constructed. The probability is 0.1
- because out of ten digits (0, 1, 2, ...., 9} each digit has equal probability i.e., 1/10 or 0.1. These are also known as random digits.
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- The simplest ways of achieving such numbers are games of chance such as dice, coins, cards or by repeatedly drawing numbered slips out of a hat. These are usually grouped purely for convenience of reading but this would become very tedious for long runs of each digits. Fortunately tables of random digits are no w widely available (see table 7.1).
- For implementation on computers to provide sequences of such digits easily and quickly, the most common methods are called Pseudo random techniques. Here, digits will re-appear in the same order (i.e., cycle) eventually but for a good technique the cycle might be tens of thousands of digits long. Of course the Pseudo random digits as the title says, are not truly random. In fact, they are completely deterministic but they do exhibit most of the properties of random digits.