3: Probability Distributions and Statistics
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- 3.1: Random Variables
- A random variable is a number generated by a random experiment. A random variable is called discrete if its possible values form a finite or countable set. A random variable is called continuous if its possible values contain a whole interval of numbers.
- 3.2: Probability Distributions for Discrete Random Variables
- The probability distribution of a discrete random variable X is a list of each possible value of X together with the probability that X takes that value in one trial of the experiment. The probabilities in the probability distribution of a random variable X must satisfy the following two conditions: Each probability P(x) must be between 0 and 1 and the sum of all the probabilities is 1 .
- 3.3: The Binomial Distribution
- Suppose a random experiment has the following characteristics. There are n identical and independent trials of a common procedure. There are exactly two possible outcomes for each trial, one termed “success” and the other “failure.” The probability of success on any one trial is the same number p. Then the discrete random variable X that counts the number of successes in the n trials is the binomial random variable with parameters n and p. We also say that X has a binomial distribution
- 3.4: Continuous Random Variables
- For a discrete random variable X the probability that X assumes one of its possible values on a single trial of the experiment makes good sense. This is not the case for a continuous random variable. With continuous random variables one is concerned not with the event that the variable assumes a single particular value, but with the event that the random variable assumes a value in a particular interval.
- 3.5: The Standard Normal Distribution
- A standard normal random variable Z is a normally distributed random variable with mean μ=0 and standard deviation σ=1.
- 3.6: Probability Computations for General Normal Random Variables
- Probabilities for a general normal random variable are computed after converting x-values to z-scores.