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# The Mean of a Discrete Random Variable

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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 1 2 3 4 5 6 Frequency 12 9 11 10 7 The mean is given by 1st x-value  1st frequency But, can be replaced by , the probabilities of getting 1, 2, . . . So, the mean

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

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

e.g. 1. A random variable X has the probability distribution
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.

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

E(X) = m = E(X) = A= 0 as = if x =  A16= = -26 Area =  = 26 So the mean = 26

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

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

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

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

Variance of a continuous random variable
Var(X) = 2 = This formula is similar to that used in discrete random variables Var(X) = x2P(X = x) – mean2 Replace by and P(X = x) by f(x) Ex2 From the example above E(X) = 2 So m = 2

Var(X) = 2 = A4 = A0 = 0 Area = = 4.8 So Var(X) = = = 0.8 S.D = = 0.9

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