MDM4U – Grade 12 Data Management – Continuous Probability Distributions

• Continuous Probability Distributions
• a random variable that can assume all possible random values (ie city temperature)
• Probability Density Function: a function that describes how likely this random variable will occur at a given point.
• Height formula:  height = 1/(b-a) where b is the top range, and a is the bottom range given.

• The Normal Distribution
• used to solve continuous probabilities
• total area under the curve is 1
• standard deviation is the distance from the mean to the point of inflection
• Any normal distribution can be described as by the mean and the variance: so we often write N(mean, variance) to describe a distribution
• The distribution chart shows area under the graph from the X value to the left end
• Z-Scores can be calculated using Normal distributions
• Z = x – mean / standard deviation
• Sometimes, you will have to subtract the mean to equalize. This makes it so the mean is on the center.

• Normal Approximation
• Step 1: Check if a normal approximation is appropriate. Test if np > 5 and nq > 5.
• Step 2: Estimate the mean and standard deviation (mean = nq, SD = √(npq) )
• Step 3: Estimate the probability using z-score method from above.

• Confidence Interval
• x- z * (σ/√n) < μ < x + z * (σ/√n)
• where x is the mean of the sample
• z is the z score of acceptable error
• μ is the mean of population
• n is the size of sample
• σ is the standard deviation
• Confidence levels and z- scores are retrieved from a given chart below: