4.1 random variables
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4.1 Random Variables
A random variable is a real-valued function defined on the sample space S.
π(π = π) = (1 β π)ππ, π = 1, 2, β¦ , π β 1
Suppose that there are N distinct types of coupons and that each time one obtains a coupon, it is, independently of
previous selections, equally likely to be any one of the N types. A random variable of interest is T, the number of
coupons that needs to be collected until one obtains a complete set of at least one of each type.
Solution: Fix n and define the events A1,A2, . . . ,AN as follows: Aj is the event that no type j coupon is contained
among the first n coupons collected. Then
Hence
and
Cumulative Distribution Function
Properties of F(x) [section 4.9]
5.
π{ π β€ π} = πΉ(π), π{ π < π} = πΉ(πβ) = limπ₯βπ
πΉ(π₯)
Answers: (a) 11/12 (b) 1/6 (c) ΒΎ (d) 1/12
4.2-5 Discrete Random Variables
Example. Graph of the pmf of the random variable representing the sum when two dice are rolled
Example. Suppose that the probability mass function of X is
Expected Value (or mean)
If X is a discrete random variable having a probability mass function p(x), then the expectation, or
the expected value, of X, denoted by E[X], is defined by
Β΅ =
E[X] = (-1)Γ.10 + 0Γ.25 + 1Γ.30 + 2Γ.35 = .90
Another motivation of the definition of expectation is provided by the frequency
interpretation of probabilities. Think of X as representing our winnings in a single game of
chance. That is, with probability p(xi) we shall win xi units i = 1, 2, . . . , n. By the
frequency interpretation, if we play this game continually, then the proportion of time
that we win xi will be p(xi). Since this is true for all i, i = 1, 2, . . . , n, it follows that our
average winnings per game will be
β π₯ππ(π₯π) = πΈ[π]
π
π=1
Expectation of a function of a random variable
If X is a random variable, then for any real function g defined on the range of X, Y = g(X) is a random variable.
πΈ[π] = β π¦ππ(π = π¦π)
π
Solution 1: Solution 2:
A computation on page 130 shows that
Variance
Here Β΅ = E(X). From Proposition 4.1 it follows that a discrete random variable
Important property:
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