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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...
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...
Probability and Statistics II week 9
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- Class notes
- • 12 pages •
Probability and Statistics II week 10
THE BETA DISTRIBUTION X is uniform on the interval [a, b], the p.d.f. of the uniform distribution is defined as: The uniform density is a special case of the beta distribution, which is defined in the following way: Definition 3.2.5. A random variable X has a beta distribution and it is referred to as a beta random variable if and only if its probability density function is given by where are the parameters of the distribution. In recent years, the beta distribution has found ...
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- Class notes
- • 17 pages •
THE BETA DISTRIBUTION X is uniform on the interval [a, b], the p.d.f. of the uniform distribution is defined as: The uniform density is a special case of the beta distribution, which is defined in the following way: Definition 3.2.5. A random variable X has a beta distribution and it is referred to as a beta random variable if and only if its probability density function is given by where are the parameters of the distribution. In recent years, the beta distribution has found ...
Probability and Statistics II week 8
1.1.1. GEOMETRIC (PASCAL) DISTRIBUTION A random variable X is defined to have a geometric distribution, if the discrete probability distribution function (p.d.f) of X is given by 0<p<1. p = probability of success and q is the probability of failure. This distribution occurs in real life, in situations where we have say a random variable X representing the number of trials required before the first success, then X has a geometric distribution. The Mean, Variance and Moment generating...
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- Class notes
- • 20 pages •
1.1.1. GEOMETRIC (PASCAL) DISTRIBUTION A random variable X is defined to have a geometric distribution, if the discrete probability distribution function (p.d.f) of X is given by 0<p<1. p = probability of success and q is the probability of failure. This distribution occurs in real life, in situations where we have say a random variable X representing the number of trials required before the first success, then X has a geometric distribution. The Mean, Variance and Moment generating...
Probability and Statistics II week 7
Example 3.1 Given , p = probability of success and q is the probability of failure. Show that f (x) is a discrete probability density function (p. d. f). Solution To show that a function is a discrete p. d. f. we have to show that its sum is unity, i.e., But from the binomial formula; Hence f (x) is a p. d. f. A random variable X that has a p.d.f. of the form is said to have a binomial distribution, and any such is called a binomial p.d.f. A binomial distribut...
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- Class notes
- • 20 pages •
Example 3.1 Given , p = probability of success and q is the probability of failure. Show that f (x) is a discrete probability density function (p. d. f). Solution To show that a function is a discrete p. d. f. we have to show that its sum is unity, i.e., But from the binomial formula; Hence f (x) is a p. d. f. A random variable X that has a p.d.f. of the form is said to have a binomial distribution, and any such is called a binomial p.d.f. A binomial distribut...