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- Definition:
Random Variable
A variable the value of which is a number determined by the outcome of an
experiment.
- Example: We flip a coin. Let X = 1 if Heads, and
let X = 0 if Tails.
- Example: We roll a die. Let X = "number of dots".
- Definition:
Probability Distribution:
A function that assigns probabilities to the values of a random variable.
- Example: We flip a coin. Given the definition of a random variable
in (2), let the probability distribution be:
æ1/2 x = 0
ç
f(x) = ç1/2 x = 1
ç
è 0 otherwise
A plot of this function looks like this:
- Note that it is essential that the function, f(x), be defined
for all possible values of the argument, x -- in this case, the entire real
line. Consequently, the "0 otherwise" is not trivial!
- Example: We roll two dice. Let X = "sum of the two faces" and
let the probability distribution be:
æ1/36 x = 2 or x = 12
ç
ç2/36 x = 3 or x = 11
ç
ç3/36 x = 4 or x = 10
ç
f(x) = ç4/36 x = 5 or x = 9
ç
ç5/36 x = 6 or x = 8
ç
ç6/36 x = 7
ç
è 0 otherwise
- Properties of Discrete Probability Distributions
a) f(x_{i}) ³ 0 for all i
b) å_{i=1,n} f(x_{i}) = 1
c) P[a £ X £ b] =
å f(x_{i})
- Example: With respect to (8):
P[6 £ X £ 9] =
P[X = 6] + P[X = 7] + P[X = 8] + P[X = 9] =
1/36(5 + 6 + 5 + 4) = 20/36 = 5/9
Note that: P[6 < X £ 9] =
P[7 £ X £ 9]
- Definition: Distribution Function
F(x) = P(X £ x)
- Example: With respect to (8):
F(5) = P[X = 1] + P[X = 2] + P[X = 3] + P[X = 4] + P[X = 5] =
1/36(1 + 2 + 3 + 4) = 10/36 = 5/18
- Note that, when working with discrete distributions that:
P[X < 5] = F(4)
P[5 £ X £ 8] = F(8) - F(4)
P[5 £ X < 8] = F(7) - F(4)
P[5 < X < 8] = F(7) - F(5)
- Discrete Uniform Distribution
æ1/n x = 1,2,3,4,5,...,n
f(x) = ç
è 0 otherwise
A graph of the function looks like this:
- Examples of the Discrete Uniform Distribution
P[X £ 3] = 3/n
P[4 £ X £ n] =
1 - P[X < 4] = 1 - F(3) = (n-3)/n
F(5) = 5/n
- Bernoulli Distribution
A Bernoulli experiment or trial is an experiment with only two
possible outcomes. Traditionally these are known as a success and a
failure. Let X = 1 be a success, and X = 0 be a
failure, and let p be the probability: P(X = 1). Hence:
æ p x = 1
ç
f(x) = ç1-p x = 0
ç
è 0 otherwise
- Binomial Distribution
Let X_{1}, X_{2}, X_{3}, ..., X_{n} be random variables
corresponding to n Bernoulli trials (experiments), and let Z be the
random variable:
Z = å_{i=1,n} X_{i}
The distribution of Z is the binomial
æ ænö
ç ç ÷ p^{z}(1-p)^{(n-z)}
f(z) = ç èzø z=0,1,2,3,...,n
ç
è 0 otherwise
- Example: Let n = 4 and p = 1/2, then
P[Z = 2] = (4 choose 2)(1/2)^{2}(1/2)^{2} = 6/16 = 3/8
- Problem 3.1 p.79. Let A and B be the two impurities.
We are given that P(A) = .4, P(B)= .5, and
1 - P(A È B) = .2. Hence
P(A È B) = .8 and
P(A È B) = P(A) + P(B) -
P(A Ç B) =
.4 + .5 - P(A Ç B) = .8. Hence
P(A Ç B) = .1
Let Y = number of impurites.
Clearly P(Y = 2) = P(A Ç B) = .1
and P(Y = 0) =
1 - P(A È B) = .2. Hence
æ .2 y = 0
ç
ç .7 y = 1
f(y) = ç
ç .1 y = 2
ç
è 0 otherwise
To see that P(Y = 1) = .7, note that the shaded areas below correspond
to this probability.
Hence, P(Y = 1) = P(A È B) -
P(A Ç B) = .8 - .1 = .7
- Problem 3.3 p.79
Let D = "Defective". Let Y be a random
variable and let it equal the number of the test on which the 2^{nd}
defective is found. Clearly
P(Y = 2) = P(D on draw 1)* P(D on draw 2) = 1/2 * 1/3 = 1/6
P(Y = 3) =
P(D on draw 1) * P(D^{c} on draw 2) * P(D on draw 3) +
P(D^{c} on draw 2) * P(D on draw 1) * P(D on draw 3)
Hence: P(Y = 2) = 1/2*2/3*1/2 + 1/2*2/3*1/2 = 2/6
Using similar reasoning:
P(Y = 4) = 1/2*2/3*1/2*1 + 1/2*2/3*1/2*1 + 1/2*1/3*1*1 = 3/6
æ 1/6 y = 2
ç
ç 2/6 y = 3
f(y) = ç
ç 3/6 y = 4
ç
è 0 otherwise