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week06.ppt3-8 Binomial Distribution
- A trial with only two possible outcomes is used so
frequently as a building block of a random experiment
that it is called a Bernoulli trial.
- It is usually assumed that the trials that constitute the
random experiment are independent. This implies that
the outcome from one trial has no effect on the
outcome to be obtained from any other trial.
- Furthermore, it is often reasonable to assume that the
probability of a success on each trial is constant.
3-8 Binomial Distribution
- Consider the following random experiments and random variables.
- Flip a coin 10 times. Let X = the number of heads obtained.
- Of all bits transmitted through a digital transmission channel, 10% are received in error. Let X = the number of bits in error in the next 4 bits transmitted.
Do they meet the following criteria:
- Does the experiment consist of Bernoulli trials?
- Are the trials that constitute the random experiment are independent?
- Is probability of a success on each trial is constant?
3-8 Binomial Distribution
3-8 Binomial Distribution
3-8 Binomial Distribution
3-9 Poisson Process
3-9 Poisson Process
3-9.1 Poisson Distribution
3-9 Poisson Process
3-9.1 Poisson Distribution
3-9 Poisson Process
3-9.1 Poisson Distribution
3-9 Poisson Process
3-9.1 Poisson Distribution
3-9 Poisson Process
3-9.1 Poisson Distribution
3-9 Poisson Process
3-9.2 Exponential Distribution
- The discussion of the Poisson distribution defined a random variable to be the number of flaws along a length of copper wire. The distance between flaws is another random variable that is often of interest.
- Let the random variable X denote the length from any starting point on the wire until a flaw is detected.
- As you might expect, the distribution of X can be obtained from knowledge of the distribution of the number of flaws. The key to the relationship is the following concept:
The distance to the first flaw exceeds 3 millimeters if and only if there are no flaws within a length of 3 millimeters—simple, but sufficient for an analysis of the distribution of X.
3-9 Poisson Process
3-9.2 Exponential Distribution
3-9 Poisson Process
3-9.2 Exponential Distribution
3-9 Poisson Process
3-9.2 Exponential Distribution
3-9 Poisson Process
3-9.2 Exponential Distribution
- The exponential distribution is often used in reliability studies as the
model for the time until failure of a device.
- For example, the lifetime of a semiconductor chip might be modeled
as an exponential random variable with a mean of 40,000 hours. The
lack of memory property of the exponential distribution implies
that the device does not wear out. The lifetime of a device with
failures caused by random shocks might be appropriately modeled as
an exponential random variable.
- However, the lifetime of a device that suffers slow mechanical wear,
such as bearing wear, is better modeled by a distribution that does
not lack memory.
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Poisson
