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Chapter 4 Random Variables and Probability Distributions Mohamed Elhusseiny [email protected]il.com Random Variables • In the previous chapter we dealt with Set Algebra to represent random experiments and their sample spaces. In this chapter we will deal with number and function algebra to consider these experiments and to consider what is called probability distributions • A random variable is a function for which each element of the sample space is influenced by the function X (random Variable) and assigned a unique number X: S R Random Variables X: S R • Example: S={ HH, HT, TH, TT}, let X be the random variable representing the number of heads that will appear S={ HH, HT, TH, TT}, X=2 X=1 X=0 X=1 X 0 1 2 Total P(X) 1/4 2/4 1/4 1 Probability distribution table Random Variables • Example: S={ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }, let X be the random variable representing the number of heads that will appear { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT } X=3 X=1 X=2 X=0 X 0 1 2 3 Total P(X) 1/8 3/8 3/8 1/8 1 Probability distribution table Random Variables • Example: tossing a die twice S={ (1,1), (1,2), …………………….., (6,5), (6,6)}, let X be the random variable representing the sum of the two tosses X 2 3 4 5 6 7 8 9 10 11 12 Total P(X) 1/ 2/ 3/ 4/ 5/ 6/ 5/ 4/ 3/ 2/ 1/ 1 36 36 36 36 36 36 36 36 36 36 36 Probability distribution table (1,4), (2,3), (3,2), (4,1) Types of Variables The variables used by statistics is usually classified based on there types. There are two types of variables Variables Discrete Continuous Type of Data (Qualitative) Type of Data (Quantitative) Nominal Data Ordinal Data Numerical Data Normal Distribution • Normal Distribution • The normal distribution is the most important specific continuous distribution. Given a problem involving a normal distribution, you should begin by clearly defining the relevant normal variable X. • You should next sketch a graph of the normal distribution and label it with the given information concerning the mean, standard deviation, and probabilities. • Every thing concerning normal distribution family is transformed into the standard normal random variable Z. Normal Distribution Normal Distribution • the following probabilities, where Z is the standard normal random variable: Examples Examples Examples Examples • Consider an investment whose profit is normally distributed with mean of 10% and a standard deviation of 5%. Determine the probability of losing money. ii. Find the probability that the investment will gain 15% of the money. iii. Find the probability that the investment will gain more than 15% of the money. i. Examples i. Let X be a random variable representing the profit of the investment. i) P( losing money) = P ( X < 0) X 0 10 P( ) P ( Z 2) 5 P ( Z 2) 0.5 P (0 Z 2) = 0.5 – 0.4772 = 0.0228 Examples ii. P( gain 15% of the money) = P ( X = 15) = 0 iii. P( gain more than 15% of the money) = P ( X > 15) P( X 15 10 ) P ( Z 1) 5 0.5 P (0 Z 1) = 0.5 – 0.3413 = 0.1587