2. The sum of all the probabilities in the sample space must add up to 1

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1 Continuous Random Variables and Continuous Probability Distributions Continuous Random Variable: A variable X that can take values on an interval; key feature remember is that the values of the variable are not countable like discrete random variable. For continuous random variables, the probability is calculated by finding the area under the probability distribution curve/graph. However, the conditions for probability distribution for proabilities associated with continuous random variable is the same as it is for discrete random variable, that is: 1. Individual probabilities must be 0 x 1 2. The sum of all the probabilities in the sample space must add up to 1 For probability distributions associated with continuous random variables, the area under the curve is calculated using calculus, however we will not use calculus to find area under the curve. We will use regular geometric formulas or table to find the probabilities. Also the probability function associated with a continuous random variable is called probabilty density function or density curve. Here are some interesting facts about continuous random variable X: 1. You do not find probabilities for individual X values as you did in the dicrete cases such as, Binomial and Poissom distributions, but rather you find the probability over an interval such as P(a < x < b) 2. The probability for a single point is always zero that is P(x = a) = 0 3. Since the probability of a single point is always zero for a continuous random variable P(a < x < b) = P(a x b) = P(a < x b) = P(a x < b) 1

2 Uniform Distribution A continuous random variable X has uniform distribution on a closed interval [a, b] if it satisfies: 1 f (x) = b a if a x b Note that for the uniform distribution, the probability is constant over the entire interval 1 [a, b] which is b a, and that the area under the curve is 1 because 1 (b a) = 1. b a Hence it satifies both conditions for the above function definition to be a probability distribution. The mean for the uniform distribution is µ = a + b and the variance for the uniform 2 distribution is σ 2 (a b)2 =. Note that standard deviation is the square root of the variance. 12 Example: You school wants to test the fire alarm system during the one hour period between 10:00 a.m. and 11:00 a.m. The probability of testing the fire alarm during this period is uniformly distributed that is X U(0, 60). Answer the following: 1. On the average how many minutes can you expect to wait for the alram to ring, that is find the mean. 2. What is the variance for X U(0,60)? What is the standard deviation? 3. Find the probability that a student gets less than 2 questions right by pure guessing 4. Find the probability that you will wait 40 minutes for the alarm to go off 2

3 5. Find the probability that you have to wait more than 40 minutes 6. Find the probability that you have to wait exactly 40 minutes 7. Find the probability that you will wait between 20 and 50 minutes for the alarm to go off 3

4 Exponential Distribution Exponential distribution is used when we are expecting some event to occur at some point in time in the future. It is mostly used in reliablity test such as the time until a product lasts or something similar. Here is the definition of expomnential function: f (x;λ) = { λe λx if x 0 Here λ is positive, that is λ > 0, is called the rate parameter. However in statistics we define exponential distribution slightly differently for our convenience to extract mean, standard deviation, variance of the probability density function (pdf) or density curve. 1 f (x;β) = β e x/β if x 0 Here β is called the suvival parameter and X is the continuous random variable that represents the time by which a mechanical system breaks down, such as your battery on your smart phone malfunctions or your computer hard drive malfunctions or an item that you purchased does not work properly anymore. The distribution is denoted by X Exp(β) Note the relationship between β and λ above which is λ = 1. Interesting observations: the β mean of the exponential distribution is β, the standard deviation is β, variance is β 2, and the mode is βln2 The cumulative probability for exponential distribution is calculated by: P(X x) = { 1 e x/β if x 0 In case if you wanted to find the cumulative probability in terms of the parameter λ then the formula becomes: P(X x) = { 1 e λx if x 0 4

5 Example: Your lifeline!!! your favorite iphone has average battery life β = 7 years and has exponential distribution X Exp(β). 1. Find the probability that your favorite iphone battery will last for at least 4 years. 2. After 4 years your phone is still working great, but you want to sell it and buy a new one. But the new buyer wants you to tell her the probabilty that the phone battery will last for another 3 years. Can you tell her the probability. Example: The average life expectency in the U.S. is about 80 years (to clarify, it is rather years as of 2015 and for Japan it is about 84 years, the highest in the world). The life expectancy X for a person living in the U.S. is X Exp(β). 1. Find the probability that an individual living in the U.S. will live at least 75 years. 2. Find the probility that a person will live less than 45 years 3. Find the probility that a person will be a centurian 4. Find the probility that a person will live between 45 years 85 years 5

6 Normal Distribution The normal distribution is also known as the Gaussian distribution. The Gaussian/normal distribution is defined as: f (x) = 1 σ 2 /2σ 2π e (x µ)2 Here are some properties of normal distribution: 1. The values of x comes from the real number line whihc means < x < 2. The distribution is bell shaped and symmetric 3. The mean µ is a parameter that is the center of the sata points and the variance σ 2 is the spread of the data points 4. The empirical rule is derived from the bell shaped curve or Gaussian/normal distribution % 95% Frequency 99.7% 2.1% 13.6% 34.1% 34.1% 13.6% 2.1% Standard deviations 6