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Mathematics LibreTexts

6.4: Binomial distribution and Normal Distribution

  • Page ID
    7330
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    Discrete probability distributions

    Binomial Trials

    1. There are a fixed number of independent trials \( n. \)
    2. Each trial has only two (hence binomial) outcomes, either “success” or “failure”.
    3. For the trials, the probability of success,\( p\) is always the same, and the probability of failure, \(q=1- p\), is also always the same.
    4. The expected value \( E(X)= np\).

    Excel Activity

    Goal: Get a “feel” for binomial distributions by finding their probability distribution tables and graphing them.

    Calculate the probability distribution table for X, a binomial distribution with 10 trials and probability of success p = 0.02. Use the drag feature to save yourself from a lot of typing!

    X

    P(X = x)

    0 (say this is in cell A2)

    =BINOMDIST (A2, 10,0.2,False)

    1

     

    2

     
       
       
       
       

    Use Chart Wizard to plot the probabilities as a histogram (bar chart with no gaps!) You’ll need to click on the bars of the chart and Select Data to get the 0, 1, 2, … as the X-axis labels and you’ll need to Select Format Data Series to remove gaps.

    Repeat for n=10, p= 0.5 and n=10, p = 0.9. You’ll get 3 tables and 3 histograms. What are the shapes of each distribution?

    Answer the following: For small n Binomial Histograms tend to be ______ skewed if p < 0.5 and ______ skewed if p > 0.5.

    Continuous probability distribution