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Probability and Statistics II week 14
BIVARIATE EXPECTATION There are many problems in which we are interested not only in the expected value of a random variable X, but also in the expected values of random variables X and Y. Theorem 1 If X and Y are discrete random variables and is the value of their joint probability distribution at , the expected value of is Correspondingly, if X and Y are continuous random variables and is the value of their joint probability density at , the expected value of is Example 13 W...
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- • 27 pages •
BIVARIATE EXPECTATION There are many problems in which we are interested not only in the expected value of a random variable X, but also in the expected values of random variables X and Y. Theorem 1 If X and Y are discrete random variables and is the value of their joint probability distribution at , the expected value of is Correspondingly, if X and Y are continuous random variables and is the value of their joint probability density at , the expected value of is Example 13 W...
Probability and Statistics II week 13
MARGINAL DISTRIBUTIONS Definition 5 If X and Y are discrete random variables and is the value of their joint probability distribution at , the function given by: for each x within the range of X is called the marginal probability distribution function of X. Correspondingly the function given by: for each y within the range of Y is called the marginal probability distribution function of Y. Example 7 The joint probability distribution function of random variables X and Y are given in t...
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- • 22 pages •
MARGINAL DISTRIBUTIONS Definition 5 If X and Y are discrete random variables and is the value of their joint probability distribution at , the function given by: for each x within the range of X is called the marginal probability distribution function of X. Correspondingly the function given by: for each y within the range of Y is called the marginal probability distribution function of Y. Example 7 The joint probability distribution function of random variables X and Y are given in t...
Probability and Statistics II
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads i
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CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads i
Probability and Statistics II week 12
CHAPTER FOUR: BIVARIATE DISTRIBUTION THEORY We earlier defined random variable as a real-valued function over a sample space with a probability measure, and it stands to reason that many different random variables can be defined over one and the same sample space. In this section we shall be concerned with the bivariate case, i.e. with situation where we are interested at the same time in a pair of random variables defined over a joint sample space. If X and Yare discrete random variables, w...
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- • 13 pages •
CHAPTER FOUR: BIVARIATE DISTRIBUTION THEORY We earlier defined random variable as a real-valued function over a sample space with a probability measure, and it stands to reason that many different random variables can be defined over one and the same sample space. In this section we shall be concerned with the bivariate case, i.e. with situation where we are interested at the same time in a pair of random variables defined over a joint sample space. If X and Yare discrete random variables, w...
Probability and Statistics II week 11
From Equation (3.2.3), we see that the normal distribution has the cumulative distribution function , ……………………………………………(3.2.4) From this we obtain , ………………………………(3.2.5) The integral in (3.2.4) cannot be evaluated by elementary methods, but can be represented in terms of the integral ……………………………………………………………………(3.2.6) which is the distribution function of the normal distri...
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- • 15 pages •
From Equation (3.2.3), we see that the normal distribution has the cumulative distribution function , ……………………………………………(3.2.4) From this we obtain , ………………………………(3.2.5) The integral in (3.2.4) cannot be evaluated by elementary methods, but can be represented in terms of the integral ……………………………………………………………………(3.2.6) which is the distribution function of the normal distri...
Probability and Statistics II week 2
There are many problems in which it is of interest to know the probability that the value of a random variable is less than or equal to some real number x. Thus, let us write the probability that X takes on a value less than or equal to x as and refer to this function defined for all real numbers x as the distribution function, or the cumulative distribution function, of X Definition 1.3 (Distribution Function or Cumulative Distribution Function) If X is a discrete random variable, the fu...
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- • 14 pages •
There are many problems in which it is of interest to know the probability that the value of a random variable is less than or equal to some real number x. Thus, let us write the probability that X takes on a value less than or equal to x as and refer to this function defined for all real numbers x as the distribution function, or the cumulative distribution function, of X Definition 1.3 (Distribution Function or Cumulative Distribution Function) If X is a discrete random variable, the fu...
Probability and Statistics II Week 1
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
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- • 8 pages •
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
Probability and Statistics Week 1
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
- Class notes
- • 8 pages •
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
Probability and Statistics Week 1
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
- Class notes
- • 8 pages •
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
Probability and Statistics Week 1
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...
- Class notes
- • 8 pages •
CHAPTER ONE 1.1. RANDOM VARIABLES One of the basic ideas in probability is that of a random variable. In many cases this is simply the numerical variable under consideration. For example, if a coin is tossed twice, the number of heads which turn up can be either 0, 1 or 2 according to the outcome of the experiment. We say that the number of heads is a random variable, since it expresses the result of the experiment as a number. Other simple random variables are the temperature of a chemical pr...