Random Variables and Sampling Distribution, Cheat Sheet of Global studies

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Random Variables

(1) Random Variables (2) Discrete Random Variable (4) Random Variable Characteristics

Random Variables

Random Variables Random Variables are ways to map outcomes of random processes to numbers (Quantifying the outcomes). Random Processes like:

• Flipping a coin
• Rolling Dice
• Measuring The Rain that might fall tomorrow A random variable is a function that associates a real number with each element in sample space. Discrete Random Variable If the range of a random variable is a countable set (finite or infinte) the R.V. is said to be discrete Random variable.

P.M.F.

(1) Probability Mass Function (P.M.F.)

P.M.F.

Probability Mass Function (P.M.F.) f(x) = p (X=x) denotes the probability that x random variable x takes the value of x PMF must satisfy: 1 - f(x) ≥ 0 2 -

C.D.F.

Probability Mass Function (P.M.F.) f(x) = p(X≤x) = denotes the probability that x random variable x takes the value of x CDF must satisfy: 1 - Non-decreasing function/prop 2 - F(-∞) = zero 3 - F(∞) = 1 & F:R → [0,1]

Expectations

(1) Expectation - E(x) (2) Properties of Expectations

Variance

(1) Variance - V(x) (2) Properties of Variance ( 3 ) Standard Deviation - (4) Measurements

Variance

Variance - V(x) 2 )– [E(x)] 2 = E(x 2 V(x) = E(x-μ) It measures the dispersion (variability) in the distribution. Properties of Variance (1) (2) (3) Standard Deviation -

Sampling Distribution

(1) Population vs Sample (2) Statistics Laws ( 3 ) Some Laws (4) Conditions of a Random Sample (5) The Distribution of 𝒙̅

Sampling Distribution

Population vs Sample It measures the dispersion (variability) in the distribution. Population Sample a set of all elements under subset of population investigation

• much easier
• less time
• less cost
• hard to define
• hard to observe
• hard to contact (Statistics) 1 - Sample Mean (𝑥̅ ) ) 2 2 - Sample Variance (S 3 - Sample std.deviatiation (S) (Parameters) 1 - Population Mean (μ) 2 - Population Variance (𝜎) 3 - Population std.deviatiation (𝜎 𝟐 )

Sampling Distribution

Conditions of a Random Sample 1 - x 1 , x 2 , …… xn are independent f(x 1 , x 2 , ……, xn) = f(x 1 ) f(x 2 ) …… f(xn ) Selection of Xidoesn’t affect selection of X i+ 2 - x 1 , x 2 , x 3 , …… xn have the same distribution They are identically distributed E(x 1 ) = E(x 2 ) = E(x 3 ) = …… = E(xn ) V(x 1 ) = V(x 2 ) = V(x 3 ) = …… = V(xn )

Sampling Distribution

The Distribution of 𝒙̅ (Case 1) If: x 1 , x 2 , ……, xn is a RS from normal population with mean and variance If: population is normal Then 𝑥̅ is normal (μ, 𝜎 2 𝑛

Any random sample from normal distribution normal (μ, 𝜎 2 ) have sample mean where 𝑥̅ follows normal (μ, 𝜎 2 𝑛

Sampling Distribution

(Case 2) Central Limit Theorem If: x1, x2, ……, xn is a random sample from any population with mean and variance Then: for large enough n (n≥30) 𝑥̅ follows normal (μ, 𝜎 2 𝑛

follows n(0,1) as n ≥ 30

Case of 2 Populations

(1) Case of 2 Populations (2) Some Laws