Continuous Probability Distribution By: Dr. Wan Azlinda Binti Wan Mohamed Continuous Random Variable A continuous random variable is one that can assume an uncountable number of values. We cannot list the possible values because there is an infinite number of them. Because there is an infinite number of values, the probability of each individual value is virtually 0. Consequently, we can only determine the probability range of values. Normal Distribution The normal distribution is the most important of all probability distribution because of its crucial role in statistical inference Examples of continuous variables are the height of adult men, body temperatures of rats, and cholesterol levels of adult man. Many of continuous variables, have distributions that are bell-shaped and are called approximately normally distributed variables. Histograms for the Distribution of Heights of Adult Women Normal Distribution The distribution is also known as the bell curve or the Gaussian distribution, named for the German mathematician Carl Friedrich Gauss ( ), who derived its equation. Normal Probability Density Function The normal density function of a normal random variable is where e = and = Normal Distribution The area under the normal distribution curve is more important than the frequencies The shape and position of the normal distribution curve depends on two parameters, the mean () and the standard deviation ( ) Normal Distribution Different mean but same standard deviation Same mean but different standard deviation Different mean and different standard deviations Summary of the properties of the Theoretical Normal Distribution The normal distribution curve is bell shaped The mean, median, and mode are equal and located at the center of distribution The normal distribution curve is unimodal (i.e. it has only one mode) The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of the vertical line passing through the center. Summary of the properties of the Theoretical Normal Distribution The curve is continuous- i.e there is no gaps or holes. For each value of X, there is a corresponding value of Y. The curve never touches the x-axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis- but it gets increasingly closer. Summary of the properties of the Theoretical Normal Distribution The total area under the normal distribution curve is equal to 1.00, or 100%. This fact may seem unusual, since the curve never touches the x-axis, but one can prove it mathematically by using calculus. The area under the normal curve that lies within one standard deviation of the mean is approximately 0.68, or 68%; within two standard deviations, about 0.95, or 95%; and within three standard deviations, about 0.997, or 99.7%. Area under the normal curve The Standard Normal Distribution The Standard Normal distribution is a normal distribution with mean of 0 and standard deviation of 1. All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score: z = value mean or standard deviation z = X The normal distribution curve can be used as a probability distribution curve for normally distributed variables. The area under the curve correspond to the probability. Finding the area under the Normal Distribution Curve Example: Find the area under the normal distribution between 1. z = 0 and z = z = 0 and z = 0 and z = -1.75 Solution: Find the area under the normal distribution between 1. z = 0 and z = 2.34 ( OR 49.04%) 2. z = 0 and 1.8 ( OR %) 3. Z = 0 and z = ( OR 45.99%) Exercise: Find the area 1. to the right of z = to the left of z= between z = 2.00 and z= between z = and z = between z = and z = to the left of z = to the right of z = to the right of z = and to the left of z= -3.01 Solution: Find the area 1. to the right of z = 1.11 ( or 13.35%) 2. to the left of z= ( or 2.68%) 3. between z = 2.00 and z= 2.47 ( or 1.60 %) 4. between z = and z = ( or 19.67%) 5. between z = and z = -1.37( or86.82%) 6. to the left of z = 1.99 ( or %) 7. to the right of z = ( or %) 8. to the right of z = and to the left of z= ( or 0.88 %) Example: Find the probability for each 1. P(0 < z < 2.32) 2. P (z < 1.65) 3. P ( z> 1.91) Example: Find the probability for each 1. P(0 < z < 2.32) ( r 48.98%) 2. P (z < 1.65) ( or %) 3. P ( z> 1.91) ( or 2.81 %) Example: Find z values such that the area under the normal distribution curve between 0 and the z value is Application of Normal distribution To solve problems by using the standard normal distribution, transform the original variable into a standard normal distribution by using the formula: z = value mean or standard deviation z = X Example: If the score for the test have a mean 100 and a standard deviation of 15, find the percentage of scores that will fall below 112 Example: Each month, a typical University office generates an average of 28 pounds of paper for recycling. Assume that the standard deviation is two pounds. If University office is selected at random, find the probability of its generating a. Between 27 and 31 pounds per month b. More than 30.2 pounds per month. Assume the variable is approximately normally distributed. Example: The University Help Line reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. If 80 calls are randomly selected, approximately how many will be responded to in less than 15 minutes. Example In order to qualify for the University scholarship, candidate must score in the top 10% of a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify. Assume the test scores are normally distributed. Example: For a medical study, a researcher wishes to select people in the middle 60 % of population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in the study. Thank You