Download Discrete Probability Distributions

Discrete Probability Distributions
• Discrete Probability Distributions have 3 major
properties:
• 1) ∑ P(X) = 1
• 2) P(X) ≥ 0
• 3) When you substitute the random variable into
the function, you find out the probability that the
particular value will occur.
• Three major probability distributions: Binomial
distribution, Hypergeometric distribution, Poisson
distribution.
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The expected value or mean of a discrete
random variable
For a discrete random variable X with probability
mass function
P(X = x) = p(x)
over a specified range, the mean of X or the
expected value of X or the expectation of X is
given by
where the sum is taken over the range of X
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Example
The probability distribution for the random variable D, the
number on the face of a die after a single toss:
D
1
2
3
4
5
6
P(D)
1/6
1/6
1/6
1/6
1/6
1/6
DP(D)
1/6
2/6
3/6
4/6
5/6
6/6
μ = E(X) = 21/6 = 3.5
The expected value is a single average value that summarizes a
probability distribution. On average, the value you expect
from a toss of a die is 3.5. This is the population mean.
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HW
•
Find E(x)
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x=0,1,2,3,4
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Continuous Random Variables
• A nondiscrete random variable X is said to be
absolutely continuous, or simply continuous, if its
distribution function may be represented as
x
F(x) = P(X ≤ x) =−∞∫ f(u) du
where the function f(x) has the properties
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The mean of a continuous random
variable
For a continuous random variable X with
probability density function f (X) over a
specified range, the mean of X or the
expected value of X is given by
where the integral is taken over the range of X.
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Example
The p.d.f. of the continuous random variable T ~ Triangular(20)
is given by
,the mean of T is
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Mean of a Linear Function of a Random
Variable
Let X be a random variable and let
Y = aX + b,
where a and b are given scalars. Then,
E[Y ] = aE[X] + b,
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