Probability of distrbutions

Question 1

Random variable - 2

Answer 1

• A random variable is a variable that assigns a numerical value to each outcome of a probability experiment
• X random variable, x value of random variable

Question 2

Probability distribution - 1

Answer 2

• Determines the probabilities of the possible values of a random variable

Question 3

Discrete random variable - 3

Answer 3

• Finite or countable different values.
• Probability distribution: collection of all probabilities of these values
• Sum of probabilities = 1

Question 4

Continuous random variable - 3

Answer 4

• Uncountable different values (intervals)
• Probability distribution: probability density function, probability is are under the function.
• Total are = 1

Question 5

Find probability distribution of discrete random variable - 5

Answer 5

• Determine sample space of probability experiment and probabilities of each outcome
• List numerical values X(ω) for ω ∈ Ω
• For each numerical value x of X find collection of simple events which have particular numerical value x {X = x} = {ω : X(ω) = x}
• use P({w}) determine probability of event {X = x}
  P(X =x)= P({ω : X(ω) =x})= sum(ω:X (ω)=x) P({ω}
• Tabulate results with left column with all values x of X and a column with probabilities P(X = x)

Question 6

Experiment vs random variable - 2

Answer 6

• Experiment: possible outcomes of experiment, probability of outcome
• Random variable: possible values of random variable, probability of value

Question 7

Expected value (expectation/mean) - 3

Answer 7

• X is a discrete random variable
• Expected value of X is weighted average of the possible values of X
• μ = sum(k, i = 1) xi * P(X = xi)

Question 8

Variance of X - 1

Answer 8

• σ^2 =Var(X) = sum(k, i = 1) (xi −μ)^2 2 * P (X =xi).

Question 9

Standard deviation of X - 1

Answer 9

• σ = SD(X) = √Var(X)

Question 10

Theorem Law of Large Numbers - 2

Answer 10

• Let X1,…,Xn be n independent versions of the random variable X, where X has expected value μ.
• Then the mean n1 (X1 + . . . + Xn ) of these n versions tends to approach the expected value μ.