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But,... can be replaced by,... the probabilities of getting 1, 2,... 1 st x -value  1 st frequency The Mean of a Discrete Random Variable We can find.

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Presentation on theme: "But,... can be replaced by,... the probabilities of getting 1, 2,... 1 st x -value  1 st frequency The Mean of a Discrete Random Variable We can find."— Presentation transcript:

1 But,... can be replaced by,... the probabilities of getting 1, 2,... 1 st x -value  1 st frequency The Mean of a Discrete Random Variable We can find the mean of a discrete random variable in a similar way to that used for data. Suppose we take our first example of rolling a die. Number on die 123456 Frequency 1291110711 The mean is given by So, the mean

2 When dealing with a model, we use the letter  for the mean (the greek letter m). We write or, more often, replacing p by, Notation for the Mean of a Discrete Random Variable Instead of , we can also write E(X ). This notation comes from the idea of the mean being the E xpected value of the r.v. X. pronounced “mew”

3 When dealing with a model, we use the letter  for the mean (the greek letter m). We write or, more often, replacing p by, Instead of , we can also write E(X ). ( Think of this as being what we expect to get on average ). pronounced “mew” This notation comes from the idea of the mean being the E xpected value of the r.v. X. Notation for the Mean of a Discrete Random Variable

4 e.g. 1. A random variable X has the probability distribution P x (X = ) Find (a) the mean of X. Solution: (a) mean, Tip: Always check that your value of the mean lies within the range of the given values of x. Here, or 5·25, does lie between 1 and 10.

5 Expectation of a continuous random variable This formula is similar to that used in discrete random variables E(X) =  x  P(X = x) E(X) =  = Ex1 Find the expected value of The boundaries are 16 and  as x can take any value greater than 16 E(X) =  =

6 E(X) = A  = 0 as = 0 if x =  A 16 = = -26  Area = 0 - -26  = 26  So the mean = 26  E(X) =  =

7 Symmetry If a function is symmetrical then the expected value lies on the line of symmetry Ex2 From the sketch the line of symmetry clearly lies on x = 2 So E(X) = 2

8 Variance of a Discrete Random Variable The variance of a discrete random variable is found in a similar way to the one we used for the mean. For a frequency distribution, the formula is Replacing by etc. gives

9 So, But we must replace by The variance of X is also written as Var( X ).

10 Solution: e.g. 1 Find the variance of X for the following: x P(X = x ) We first need to find the mean, 

11 Variance of a continuous random variable Var(X) =  2 = This formula is similar to that used in discrete random variables Var(X) =  x 2  P(X = x) – mean 2 From the example above E(X) = 2So  = 2 Ex2

12 A 4 = 4.8 A 0 = 0 Area = 4.8 - 0 = 4.8 So Var(X) = S.D == 0.9 = 4.8 - 2 2 = 0.8 Var(X) =  2 =


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