Overview Of Probability Distribution

Standard Distributions  Learning Objectives  Be familiar with the standard distributions (normal, binomial, and exponential).  Use standard distributions to solve for events in problems in which the distribution belongs to those families.

Example #1 Say you were buying a new bicycle for going back and forth to school. You want to buy something that lasts a long time and something with parts that will also last a long time. You research on the internet and find one brand “Buy Me Bike” that shows the following graph with all of its advertising

(a) What type of probability distribution is being represented by this graph? (b) Is the data represented continuous or discrete? How can you tell? (c) Does the data in the graph indicate that the company produces bicycles that have a respectable life span? Explain.

Let’s move into the concept of distribution…. A distribution is simply the description of the possible values of the random variables and the possible occurrences of these. For our discussions, we will say it is the probability of the occurrences. The main form of probability distribution is standard distribution. Standard distribution is a normal distribution and often people refer to it as a bell curve

Example #2 If you were to toss a fair coin 100 times, you would expect the coin to land on tails close to 50 times and heads 50 times. However, tails may not appear as expected. Look at the histograms below.

Example # 2 cont….  Notice that when we actually flipped the 100 coins in our experiment, we saw that tails come up 70 times and heads only 30 times.  The theoretical probability is what we would expect to happen. In a regular fair coin toss, we have an equal chance of getting a head or a tail. Therefore, if we flip a coin 100 times we would expect to see 50 heads and 50 tails.  When we actually flip 100 coins, we actually saw 70 tails and 30 heads. If we were to repeat this experiment, we might see 60 tails and 40 heads.

Normal Distribution  If we were to keep doing this flipping experiment, say 500 times, we may see the values get closer to the theoretical probability (the histogram on the left). As the number of data values increase, the graph of the results starts to look a bell-shaped curve.  This type of distribution of data is normal or standard distribution. The distribution of the data values is shown in this curve. The more data points, the more we see the bell shape.

Binomial Experiment What is interesting about our flipping coin example is that it is a binomial experiment. What is meant by this is that it does not have a standard distribution but a binomial distribution. Why? This is because binomial experiments only have two outcomes. Think about it. If we flip a coin, choose between true or false, choose between a Mac or a PC computer, or even asked for tea or coffee at a restaurant, these are all options that involve either one choice or another. These are all experiments that are designed where the possible outcomes are either one or the other. Binomial experiments are experiments that involve only two choices and their distributions involve a discrete number of trials of these two possible outcomes. Therefore a binomial distribution is a probability distribution of the successful trials of the binomial experiments