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4.2 – Uniform Distributions A uniform distribution occurs when, in a single trial, all outcomes are equally likely. 1 For all outcomes, 𝑥, 𝑃(𝑥) = 𝑛, where 𝑛 is the number of possible outcomes in the experiment. Example 1: Uniform Distribution A calculator has been programmed to generate a random number between 1 and 5. a) Classify this distribution. Each random number is equally likely and there is a single trial. So, this is a uniform distribution. b) Calculate the probability distribution. Random Number, x P(x) x . P(x) 1 1 5 1 5 2 1 5 2 5 3 1 5 3 5 4 1 5 4 5 5 1 5 5 5 c) Sketch a graph of the distribution. Comment on the shape of the graph. Since this is a uniform distribution and all the probabilities are equal, the bars all have the same dimensions. d) Calculate the expectation. Interpret its meaning. 1 2 3 4 5 1 1 𝐸(𝑋) = 5 + 5 + 5 + 5 + 5 = 3 OR 𝐸(𝑋) = 5 (1 + 2 + 3 + 4 + 5) = 5 (15) = 3 𝑛(𝑛 + 1) 𝑦𝑜𝑢 𝑐𝑎𝑛 𝑢𝑠𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑟𝑜𝑚 1 𝑡𝑜 𝑛 2 The expectation is 3. The predicted average value of the random number will be 3. Mathematics of Data Management (MDM4UC) Page 1 4.2 – Uniform Distributions Example 2: Fair Game A game involves rolling a die. A player who rolls an even number receives points equal to two times the face value of the die. If the player rolls an odd number, the player loses three times the face value of the die. Is this a fair game? Calculate the expectation. Points received indicate a positive value for the random variable, x. Points lost indicate a negative value for x. Roll Point value, 𝒙 𝑷(𝒙) 𝒙 ∙ 𝑷(𝒙) 1 3 1 −3 − 6 6 1 4 2 4 6 6 1 9 -9 3 − 6 6 1 8 8 4 6 6 1 15 -15 5 − 6 6 1 12 12 6 6 6 3 Sum: − 6 3 𝐸(𝑋) = ∑ 𝑥 ∙ 𝑃(𝑥) = − = −0.5 6 A fair game will have an expectation equal to 0. This is not a fair game because the player will lose 0.5 points on each turn, on average. Practice: (Page 157) #1-7 Mathematics of Data Management (MDM4UC) Page 2