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Just copy the link or share our page with them. We are just providing the link already available on the internet. The value of the integral. Properties of Deinite Integrals: Certain properties of deinite integral are useful in solving problems. Some of the often used properties are given below. Evaluate: 4. Ex- pression for mean and variance. Nobody knows what we will get whether a head or tail. But it is certain that either a head or tail will occur.
In a similar way if a dice is thrown. We may get any of the faces 1, 2, 3, 4, 5 and 6. But nobody knowns which one will occur. Experiments of this type where the outcome cannot be predicted are called "random" experiments. The word probability or chance is used commonly in day-to-day life. For example the chances of India and South Africa winning the world cup Cricket, before the start of the game are equal i.
We often say that it will rain tomorrow. Probably I will not come to function today. All these terms-chance, probable etc. In other words there is an uncertainty about the happening of the event. The term probability refers to the randomness and uncertainty. Trail and Event: Consider an experiment of throwing a coin. When tossing a coin, we may get a head H or tail T. Here tossing of a coin is a trail and getting a head or tail is an event.
From a pack of cards drawing any three cards is trail and getting a King or queen or a jack are events. Throwing of a dice is a trail and getting 1 or 2 or 3 or 4 or 5 or 6 is an event. Sample Space: The set of all possible cases of an experiment is called the sample space and is denoted by S. Mathematical Deinition of Probability: No.
Random Variable: A function X which transforms events of a random experiment into real numbers is called random variable. Example: Two coins are tossed at a time. If we take X is the number of heads appearing then HH becomes 2. Types of Random Variables: There are two types of random variables known as i Discrete random variable ii Continuous random variable Discrete random variable: If a random variable takes only a inite or a countable number of values, it is called a discrete random variable.
For example, when two coins are tossed the number of heads obtained is the random variable X. Where X assumes the values 0, 1, 2. Which is a countable set. Such a variable is called discrete random variable. Let p xi be a number associated with each xi. Continuous Random Variable: A random variable X is said to be continuous if it can take all possible values between certain lim- its.
Examples: 1. Life time of electric bulb in hours. Height, weight, temperature, etc. Deinition: Probability density function: A function f is said to be probability density function pdf of the continuous random variable X if it satisies the following condition: 1. Find the probability distribution of X when tossing a coin, when X is defined as getting a head. Solution: Let X denote getting a head.
When throwing a die what is probability of getting a 4? Hence the mathematical expectation E X of a random variable is simply the arithmetic mean. Properties of mathematical expectation: 1. Find the expected value of the random variable X has the following probability distribution.
Find its variance. The monthly demand for ladies hand bags is known to have the following distribution. Demand 1 2 3 4 5 6 Probability 0. Also obtain variance. Solution: x2 1 4 9 16 25 36 Demand x 1 2 3 4 5 6 Probability P x 0. An experiment which has two mutually disjoint outcomes is called a Bernoulli trail.
The two out- comes are usually called "success" and "failure". An experiment consisting of repeated number of Bernoulli trails is called Binomial experiment. A Binomial distribution can be used under the followiong conditions. Probability funciton of Binomial Distribution: Let X denotes the number of success in n trial satisfying binomial distribution conditions. X is a random variable which can take the values 0, 1, 2, Find n and p in the binomial distribution whose mean is 3 and variance is 2.
If the sum of mean and variance of a binomial distribution is 4. In tossing 10 coin. What is the chance of having exactly 5 heads. Ten coins are tossed simultaneously ind the probability of getting atleast seven heads. Find the chance that if a card is drawn at random from an ordinary pack, it is one of the jacks.
A bag contains 7 white and 9 red balls. Find the probability of drawing a white ball. A random variable X has the following distribution function. Ten coins are tossed simultaneously. Find the probability of getting exactly 2 heads. The mean and variance of a binomial variate X with parameters 'n' and p are 16 and 8 respectively. Eight coins are tossed simultaneously. Find the probability of getting at least ive heads. Four coins are tossed simultaneously.
Find the probability of getting i at least two heads ii at least one head. Six coins are tossed simultaneously. Find the probability of getting at most four heads. Expres- sions of mean and variance. Constants of normal dis- tribution results only. Properties of normal distribution-Simple problems using the table of standard normal distribution. Poisson distribution is a discrete distribution. Poisson distribution is a limiting case of binomial distribution under the following conditions.
The number of printing mistakes in each page of a book. The number of suicides reported in a particular city in a particular month. The number of road accidents at a particular junction per month. The number of child born blind per year in a hospital.
If the mean of the poisson distribution is 2. If the mean of the poisson distribution is 4. Find the value of the variance and standard deviation. In a poisson distribution if the variance is 2. Find the probability that out of 15 screws selected at random only 3 are defective. Solution: Let p-denote the probability of defective screw.
The probability that an electric bulb to be defective from a manufacturing unit is 0. In a box of electric bulbs find the probability that i Exactly 4 bulbs are defective ii More than 3 bulbs are defective. Solution: Let p-denote the probability of one bulb to be defective. Like poisson distribution, the normal distribution is obtained as a limiting case of Binomial distribution.
This is the most important probability model in statistical analysis. The normal curve is bell shaped. The normal curve is asymptotic to the x-axis. A normal distribution is a close approximation to the Binomial distribution when n is very large 1 and p is close to.
Standard normal distribution: A random variable Z is called a standard normal variate if its mean is zero and standard deviation is 1 and is denoted by N 0, 1. The total area under normal probability curve is unity. In the normal probability tables, the areas are given under standard normal curve.
We shall deal with the standard normal variate z rather than the variable X itself. If X is normally distributed with mean 80 and standard deviation If X is a normal variate with mean 80 and S.
If X is normally distributed with mean 6 and standard deviation 5. The mean score of students in an examination is 36 and standard deviation is If the score of the students is normally distributed how many students are expected to score more than 60 marks.
When X is normally distributed with mean 12, standard deviation is 4. In an aptitude test administered to students, the scores obtained by the students are distributed normally with mean 50 and standard deviation Find the number of students whose score is i between 30 and So we adopt the method of least squares which gives a unique set of values to the constants in the equation of the itting curve.
We apply the method of least squares to ind the value of a and b. The principle of least square consist in minimizing the sum of the square of the deviations of the actual values y from its estimated values as given by the line of best it. To obtain the line of best it. Fit a straight line to the data given below: x 0 1 2 3 4 y 1 1. Fit a straight line for the following data: Year: Production: 7 9 12 15 18 23 in tones Estimate the production for the year The variance of a poisson distribution is 0.
Give any two examples of poisson distribution. Define poisson distribution. Define normal distribution. Write any two properties of the normal curve. Free Download WordPress Themes. Download Nulled WordPress Themes. Mukhyamantri Abhyudaya Yojana Registration online. Applied mathematics second semester Loading Thanks sir Loading Okay Loading Please enter your comment!
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