Download Probability Distributions A Random Variable is a set of possible

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Probability Distributions
A Random Variable is a set of possible values from a random experiment.
They are discrete and continuous.
A discrete variable has a countable number of values (countable means values of zero, one, two, three, etc.).
A continuous variable has an infinite number of values between any two values. Continuous variables are measured.
Examples:
1. A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only take the values 0,
1, ..., 10, so X is a discrete random variable.
2. The random variable Y - height and weight. Y can take any positive real value, so Y is a continuous random variable.
The values of a random variable and their corresponding probabilities make up a probability distribution.
There are two kinds of probability distributions.
discrete (The outcomes and their corresponding probabilities can be written in a table)
Binomial distribution.
The probability of x successes is
b(n, p, x)x Cn p x (1  p) n x
EXAMPLE:
Construct a discrete probability distribution for the number of heads when three coins are tossed.
SOLUTION:
Recall that the sample space for tossing three coins is TTT, TTH, THT, HTT, HHT, HTH, THH, and HHH.
The outcomes can be arranged according to the number of heads, as shown.
0 heads TTT
1 head
TTH, THT, HTT
2 heads THH, HTH, HHT
3 heads HHH
Finally, the outcomes and corresponding probabilities can be written in a table, as shown.
Outcome, x
Probability, P(x)
0
1/ 8
1
3 /8
2
3/ 8
3
1/ 8
The sum of the probabilities of a probability distribution must be 1.
Poisson Distribution
This distribution is used when the variable occurs over a period of time, volume, area, etc. For example,
the number of white blood cells on a fixed area.
The probability of x successes is
P(n, p, x) 
e   x
x!
Where e is a mathematical constant 2.7183 and =np
A discrete probability distribution can also be shown graphically by labeling the x axis with the values of the
outcomes and letting the values on the у axis represent the probabilities for the outcomes. The graph for the
discrete probability distribution of the number of heads occurring when three coins are tossed is shown in Figure
7-1.
EXAMPLE:
1. A telemarketing company gets on average 6 orders per 1000 calls. If a company calls 500 people, find the
probability of getting 2 orders.
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The Mean and Standard Deviation for a discrete Distribution
To predict ahead of the average number of successes in n binomial trials the expected value is used. This average
can be found by using the formula mean (mu) = np where n is the number of times the experiment is repeated and p
is the probability of a success.
EXAMPLE: A die is tossed 180 times and the number of threes obtained is recorded. Find the expected number of
threes.
SOLUTION: n = 180 and p =1/6 since there is one chance in 6 to get a three on each roll.
mu = n p = 180 1/6 = 30
Hence, one would expect on average 30 threes.
Statisticians are also interested in how the results of a probability experiment vary from trial to trial. Suppose we roll
a die 180 times and record the number of threes obtained. We know that we would expect to get about 30 threes.
Now what if the experiment was repeated again and again? In this case, the number of threes obtained each time
would not always be 30 but would vary about the mean of 30. For example, we might get 28 threes one time and 34
threes the next time, etc. How can this variability be explained Statisticians use a measure called the standard
deviation. When the standard deviation of a variable is large, the individual values of the variable are spread out
from the mean of the distribution. When the standard deviation of a variable is small, the individual values of the
variable are close to the mean.
The formula for the standard deviation for a binomial distribution is standard deviation
  np(1  p)
EXAMPLE: A die is rolled 180 times. Find the standard deviation of the number of threes.
SOLUTION:
The standard deviation is 5.
Rule
-1<68%<+1
-2<95%<+2
-3<99%<+3
continuous.
The Normal Distribution
Continuous variables can be represented by formulas and graphs or curves. These curves represent probability
distributions. In order to find probabilities for values of a variable, the area under the curve between two given
values is used.
One of the most often used continuous probability distributions is called the normal probability distribution. Many
variables are approximately normally distributed and can be represented by the normal distribution. It is important to
realize that the normal distribution is a perfect theoretical mathematical curve but no real-life variable is perfectly
normally distributed.
The normal distribution has the following properties: |.
1. It is bell-shaped.
2. The mean, median, and mode are at the center of the distribution.
3. It is symmetric about the mean. (This means that it is a reflection of itself if a mean was placed
at the center.)
4. It is continuous; i.e., there are no gaps.
5. It never touches the x axis.
6. The total area under the curve is 1 or 100%.
7. About 0.68 or 68% of the area under the curve falls within one standard deviation on either
side of the mean. (Recall that ц is the symbol for the mean and a is the symbol for the standard
deviation.» About 0.95 or 95% of the area under the curve falls within two standard deviations
of the mean.
About 1.00 or 100% of the area falls within three standard deviations of the mean. (Note: It is
somewhat less than 100%, but for simplicity. 100% will be used here.)