# Continuous Probability Distributions

### Continuous probability distribution: A probability distribution in which the random variable X can take on any value (is continuous). Because there are infinite values that X could assume, the probability of X taking on any one specific value is zero. Therefore we often speak in ranges of values (p(X>0) = .50). The normal distribution is one example of a continuous distribution. The probability that X falls between two values (a and b) equals the integral (area under the curve) from a to b: # The Normal Probability Distribution

### The normal distribution is symmetric and centered on the mean (same as the median and mode).  While the x-axis ranges from negative infinity to positive infinity, nearly all of the X values fall within +/- three standard deviations of the mean (99.7% of values), while ~68% are within +/-1 standard deviation and ~95% are within +/- two standard deviations.  This is often called the three sigma rule or the 68-95-99.7 rule.  The normal density function is shown below (this formula won’t be on the diagnostic!) ### We can convert any and all normal distributions to the standard normal distribution using the equation below.  The z-score equals an X minus the population mean (μ) all divided by the standard deviation (σ). # Example Normal Problem

### We want to determine the probability that a randomly selected blue crab has a weight greater than 1 kg.  Based on previous research we assume that the distribution of weights (kg) of adult blue crabs is normally distributed with a population mean (μ) of  0.8 kg and a standard deviation (σ) of 0.3 kg.  How do we determine this probability?  First, we calculate the z score by replacing X with 1, the mean (μ) with 0.8 and standard deviation (σ) with 0.3.  We calculate our z-score to be (1-0.8)/0.3=0.6667.  We can then look in our z table to determine the p(z>0.6667) is roughly 1-0.748 (pulled from the chart, somewhere between 0.7454 and 0.7486) = 0.252.  Therefore, based on our normality assumption, we conclude that the likelihood that a randomly selected adult blue crab weighs more than one kilogram is roughly 25.2% (the area shaded in blue). 