S1 - Discrete RV's and Uniform summary? Watch

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AngelofnoColour
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Please could someone write a summary on Discrete Random Variables and Uniform distribution. Cos I always get mixed up, on which is which.

Also how do you work out the E(x), P(x<6) on both, etc...
The text books just confuse me more.
Please, thanx.
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these are crana's notes so thank her not me.

Discrete Random Variables

Random variable X
A variable that represents the values obtained when we take a measurement from an experiment in the real world.

Discrete random variable
A variable that changes by steps, and takes only specified values in any given interval.

Probability function
A function that describes how the probabilities are assigned.

Probability distribution
The set of all values of a random variable, together with their associated probabilities.

e.g.

m 1 2 3 4 5 6
P(M=m) 1/6 1/6 1/6 1/6 1/6 1/6

is the probability distribution which describes the values, m, taken when a die is thrown and the associated probabilities.

P(M=m) may also be written as P(m).

Sum of all probabilities
Σp(x) = 1

Cumulative distribution functions
F(x0) = P (X ≤ x0) = Σp(x) [for x ≤ x0]
Expected value of X
E(X) = μ = Σx P(X=x) i.e. the sum of (each value x its probability)
Expectation of a linear function of X
E(aX + b) = a E(X) + b note E(b) = b
Variance of X
Var (X) = σ2 = Σx2 P(X=x) – E(X)2
Variance of a linear function of X
Var(aX + b) = a2 Var (X) note Var(b) = 0
Expectation of a function of X
E[g(X)] = Σg(x) P (X = x)

Discrete Uniform Distribution
If X has a discrete uniform distribution:
P(X = xr) = 1/n r = 1, 2, 3 … n (must be a quantitative variable)
i.e. discrete separate values r can take
uniform each value of r has the same probability (adds up to 1)
Conditions for a discrete uniform distribution
The variable X is described over a set of n distinct (discrete) values. Each value xr is equally likely (uniform probability).
E(X) and Var(X) for a discrete uniform distribution
x = 1, 2, 3 … n
E(X) = (n + 1)/2
Var (x) = (n2 – 1)/12 or (n+1)(n – 1)/12
The Normal Distribution
Characteristics of the normal distribution
mean = median = mode (i.e. it is symmetrical)
ranges from -∞ to + ∞
is asymptotic to the x axis as x è ±∞
has a total area under the curve of unity
has notation N(μ,σ2)
The standard normal distribution
The standard normal variable is denoted by Z, where Z~N(0,12)
Any normal variable (mean μ, variance σ2) can be transformed into a standard normal variable using:
Z = x – μ remember to use σ not σ2 as denominator
σ
Φ(z) represents the area to the left of any given value z
Φ(z) = P (Z ≤ z)
If z is negative: Φ(z) = 1 - Φ(-z)
For area to the right of z: use 1 - Φ(z)
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jazzyj
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Discrete random variables are when there is a table with values of x and the corresponding probability that X=x. These are shown in a probability distribution and all the probabilities add up to 1.

To find E(X), find the sum of each of the values of x multiplied by their corresponding probability

Var(X) = E(X^2) - (E(X))^2
This means find the sum of each of the squared values of x multiplied by their probabilities minus E(X) squared

e.g. x 1 2 3
P(X=x) 0.25 0.5 0.25

E(X) = (1 x 0.25) + (2 x 0.5) + (3 x 0.25) = 2
Var(X) = {(1 x 0.25) + (4 x 0.5) + (9 x 0.25)} - 4
= 0.5
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