Download Discrete Random Variables

Discrete Random Variables
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A random variable is a function that assigns a
numerical value to each simple event in a
sample space.
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Range – the set of real numbers
Domain – a sample space from a random
experiment
A discrete random variable can assume only a
countable (finite or countably infinite) number
of values.
A continuous random variable can assume an
uncountable number of values
Counting numbers
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The values of a discrete random variable are
countable I.e. they can be paired with the
counting numbers 1,2, …
Counting numbers, 0, the negatives of
counting numbers, and the ratios of counting
numbers and their negatives (rational
numbers) are inadequate for measuring.
Consider the square root of 2, the length of
the diagonal of a square of side 1.
Measuring Numbers
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The values of a continuous random
variable are uncountable, and hence
resemble the numbers comprising a
continuum or interval, needed for
measuring
Measurements are always made to an
interval, however small.
Mass functions vs. density
functions
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With discrete random variables, probabilities
are for ‘discrete’ points
Probability functions of discrete random
variables are called probability mass functions
With continuous random variables,
probabilities are for intervals
Probability functions of continuous random
variables are called probability density
functions
Expected value of a discrete
random variable
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E(X) = S {x*[P(X=x)]}=S{x*p(x)} = m
Var(X) = S {(x-m)2 * [P(X=x)]}
= S {(x-m)2*p(x)} = s2
Laws of Expected Value
E( c ) = c
E ( cX) = cE(X)
E(X+Y) = E(X) + E(Y)
E(X - Y) = E(X) – E(Y)
E(X*Y) + E(X) * E(Y) if and only of X and
Y are independent
Laws of Variance
V(c)=0
V(cX) = c2*V(X)
V(X+c) = V(X)
V(X+Y) = V(X) + V(Y) if and only if X and
Y are independent
V(X – Y) = V(X) + V(Y) if and only if X
and Y are independent