STAT 2060 Lecture Notes - Lecture 5: Random Variable, Probability Distribution

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29 May 2017
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STAT*2060: Statistics for Business Decisions
Week 5 Lectures
1 Review from Week 4: Continuous Probability Distributions
The continuous probability distribution is represented by a probability density function, which is a
curve that describes the likelihood of all possible values of X.
The function f(x) determines the shape of this curve, with some shapes being more common than
others.
Properties of Continuous Distributions:
f(x) is the height of the curve at the point x
f(x)0
The probabilities for a continuous random variable are associated with areas under the probability
density function.
The total area under the curve is equal to 1.
With continuous random variables, we are interested in determining the probability of Xbeing in a
range of values, since it is not possible for us to find the probability of Xbeing exactly equal to a single
value.
That is, we can find:
P(X < a)
P(X > a)
P(a<X<b)
but not P(X=a)
Why not??
Since the probability associated with a continuous random variable Xis associated with the area un-
der the probability density function, and it is not possible to find the area under a single point, the
P(X=a) = 0 (that is, the probability that Xis exactly equal to a value is 0).
1
F(x)
f(x)=height of curve at point x
g(s)
f(x)
P(a less than or equal to x which is less than or equal to b)=P(a less than x less than b
Cannot measure area under one point
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2 Continuous Uniform Distribution
The simplest continuous distribution is the continuous uniform distribution, which corresponds to
a uniform random variable X.
In this distribution, the random variable Xcan take on any value between a lower bound, c, and an
upper bound, d. Within the boundaries of cto d, all intervals of equal length are equally likely to occur.
That is, within the boundaries of cto d, any interval between the points aand b, and of length l, has
an equal chance of occurring.
A consequence of this is that the shape of the continuous uniform distribution is a rectangle.
The probability that Xis between any two values, aand b, is then calculated as the area under the
curve between the points aand b. Finding this area does not require any integration, nor any statistical
tables. Since our curve is a rectangle, finding the area under the curve between the points aand bis
just finding the area of the rectangle defined by the end-points aand b, and the height of the curve f(x).
2
Area under a curve for interval of length L is same no matter where this
interval occurs since length L and height are the same
P(A<x<B)=base(height)
=(b-a) x f(x)
Height:
-use fact area=1
-base(height)
1=(d-c)f(x)
1/(d-c)=f(x)
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