Question

What is Normal, Binomial, Poisson and Exponential Distributions with examples.

What is Continuous Distributions and Density Functions.

What is Normal density and Standardizing: Z-Values

Answer 1

1)))))--->Normal Distribution:

In our probability theory, the normal distribution is a very common continuous probability distribution. The normal distribution is sometimes informally called the bell curve and is broadly utilized in statistics, business settings, and government entities such as the FDA. It's widely recognized as being a grading system for tests such as the SAT and ACT in high school or GRE for graduate students.

Example:

Formula Values:

X = Value that is being standardized

μ = Mean of the distribution

σ = Standard deviation of the distribution

-->Use the following formula to convert a raw data value, X to a standard score, Z.

-->Assuming a specific population has μ = 4, and σ = 2. For example, finding the probability of the randomly selected value being greater than 6 would resemble the following formula:

-->The Z score corresponding to X = 6 will be:

-->Z = 1 means that the value of X = 6 which is 1 standard deviation above the mean.

Poisson Distribution:

probability distribution for discrete variables is the Poisson distribution. The Poisson distribution is used to determine the probability of the number of events occurring over a specified time or space. This was named for Simeon D. Poisson, 1781 – 1840, French mathematician.

Examples of events over space or time: -number of cells in a specified volume of fluid

-->number of calls/hour to a help line

-->number of emergency room beds filled/ 24 hours

Like the binomial distribution and the normal distribution, there are many Poisson distributions.

-->Each Poisson distribution is specified by the average rate at which the event occurs.

-->The rate is notated with λ

--> λ = ‘lambda’, Greek letter ‘L’ – There is only one parameter for the Poisson distribution

The probability that there are exactly X occurrences in the specified space or time is equal to

Example:

A large urban hospital has, on average, 80 emergency department admits every Monday. What is the probability that there will be more than 100?

If we put λ =80 and x= 100 then we will get the probability value as 0.01316885.

To get the same result we can use normal approximation and then get the probability value.

emergency room admits on a Monday?

-->λ is the rate of admits / day on Monday = 80

-->we can use the normal approximation since λ > 10

The normal approximation has mean = 80 and SD = 8.94 (the square root of 80 = 8.94)

Now we can use the same way we calculate p-value for normal distribution. If you do that you will get a value of 0.01263871 which is very near to 0.01316885 what we get directly form Poisson formula. Here main intention is to show you how normal approximation works for Poisson Distribution.

Exponential Distribution:

The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.

The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.

EXAMPLE:

Let X = amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes.

X is a continuous random variable since time is measured. It is given that μ = 4 minutes. To do any calculations, you must know m, the decay parameter.

m=1/μ. Therefore, m=1/4=0.25

The standard deviation, σ, is the same as the mean. μ = σ

The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25).

The probability density function is f(x) = me^–mx. The number e = 2.71828182846… It is a number that is used often in mathematics. Scientific calculators have the key “e^x.” If you enter one for x, the calculator will display the value e.

The curve is:

f(x) = 0.25e^–0.25x where x is at least zero and m = 0.25.

For example, f(5) = 0.25e^−(0.25)(5) = 0.072. The postal clerk spends five minutes with the customers. The graph is as follows:

Notice the graph is a declining curve. When x = 0,

f(x) = 0.25e^(−0.25)(0) = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is m.

2)))--->

Continuous Distribution OR Continuous Probability Distribution

If a random variable is a continuous variable , its probability distribution is called a continuous probability distribution .

A continuous probability distribution differs from a discrete probability distribution in several ways.

-->The probability that a continuous random variable will assume a particular value is zero.

-->As a result, a continuous probability distribution cannot be expressed in tabular form.

-->Instead, an equation or formula is used to describe a continuous probability distribution.

The equation used to describe a continuous probability distribution is called a probability density function(pdf). All probability density functions satisfy the following conditions:

-->The random variable Y is a function of X; that is, y = f(x).

-->The value of y is greater than or equal to zero for all values of x.

-->The total area under the curve of the function is equal to one.

The charts below show two continuous probability distributions. The first chart shows a probability density function described by the equation y = 1 over the range of 0 to 1 and y = 0 elsewhere.

y = 1

The next chart shows a probability density function described by the equation y = 1 - 0.5x over the range of 0 to 2 and y = 0

elsewhere. The area under the curve is equal to 1 for both charts.

y = 1 - 0.5x

The probability that a continuous random variable falls in the interval between a and b is equal to the area under the pdf curve between a and b. For example, in the first chart above, the shaded area shows the probability that the random variable X will fall between 0.6 and 1.0. That probability is 0.40. And in the second chart, the shaded area shows the probability of falling between 1.0 and 2.0. That probability is 0.25.

Probability Density Functions OR Density Functions:

A continuous random variable takes on an uncountably infinite number of possible values. For a discrete random variable X that takes on a finite or countably infinite number of possible values, we determined P(X = x) for all of the possible values of X, and called it the probability mass function ("p.m.f."). For continuous random variables, as we shall soon see, the probability that X takes on any particular value x is 0. That is, finding P(X = x) for a continuous random variable X is not going to work. Instead, we'll need to find the probability that X falls in some interval (a, b), that is, we'll need to find P(a < X < b). We'll do that using a probability density function ("p.d.f."). We'll first motivate a p.d.f. with an example, and then we'll formally define it.

3))--> The Normal Density Function:

Whenever a random variable is determined by a sequence of independent random events, the outcome will be a Normal or Gaussian density function. This is known as the Central Limit Theorem. The essence of the derivation is that repeated random events are modeled as repeated convolutions of density functions, and for any finite density function will tend asymptotically to a Gaussian (or normal) function.

Standardizing: Z-Values OR Z-Score

Z-scores are expressed in terms of standard deviations from their means. Resultantly, these z-scores have a distribution with a mean of 0 and a standard deviation of 1. The formula for calculating the standard score is given below: