prob-stats e2

Random Variables

• A random variable (RV) describes the outcomes of a statistical (random) experiment in words.

• Notation:

  • Uppercase letters (X, Y) denote a random variable.

  • Lowercase letters (x, y) denote specific values of the random variable.

• Types of RVs:

  • Discrete: Countable values.

  • Continuous: Measurable values.

Discrete Random Variables

• Can assume only a countable number of values (finite or countably infinite).

• The list of all possible pairs (xi,P(xi))(x_i, P(x_i))(xi​,P(xi​)) is the discrete probability distribution function (PDF).

• Conditions:

  • Each probability must be between 0 and 1.

  • The sum of all probabilities must be 1.

Discrete Probability Distributions

• Probability Distributions: Represented as tables, graphs, or functions that assign probabilities to outcomes.

• Key properties:

  • The sum of all probabilities is 1.

  • The expected value (mean) represents the long-term average of the distribution.

  • Variance and standard deviation measure the spread of values.

Binomial Distribution

• Characteristics:

  1. A fixed number of trials (nnn).

  2. Two possible outcomes: "Success" or "Failure".

  3. Probability of success (ppp) is constant for each trial.

  4. Trials are independent.

• Formula:

  • Mean: μ=np\mu = npμ=np

  • Variance: σ2=npq\sigma^2 = npqσ2=npq

  • Standard Deviation: σ=npq\sigma = \sqrt{npq}σ=npq​

  • Probability: P(x)=(nx)pxqn−xP(x) = \binom{n}{x} p^x q^{n-x}P(x)=(xn​)pxqn−x

Geometric Distribution

• Characteristics:

  1. Bernoulli trials with all failures except the last, which is a success.

  2. The number of trials until the first success can be any positive integer.

  3. Independent trials with a constant probability of success.

• Formula:

  • Mean: μ=1p\mu = \frac{1}{p}μ=p1​

  • Variance: σ2=1−pp2\sigma^2 = \frac{1 - p}{p^2}σ2=p21−p​

  • Probability: P(X=n)=qn−1pP(X = n) = q^{n-1} pP(X=n)=qn−1p

Hypergeometric Distribution

• Characteristics:

  1. Selection of nnn items from a finite population divided into two groups.

  2. The number of items of interest in the population is rrr.

  3. Sampling is done without replacement.

• Formula:

  • Mean: μ=nrN\mu = n \frac{r}{N}μ=nNr​

  • Variance: σ2=nrNN−rNN−nN−1\sigma^2 = n \frac{r}{N} \frac{N - r}{N} \frac{N - n}{N - 1}σ2=nNr​NN−r​N−1N−n​

Poisson Distribution

• Characteristics:

  1. Counts the number of times an event occurs in a fixed interval of time or space.

  2. Events occur independently.

  3. The probability of two or more events occurring in a small interval is nearly zero.

• Formula:

  • Mean: μ=λ\mu = \lambdaμ=λ

  • Variance: σ2=λ\sigma^2 = \lambdaσ2=λ

  • Probability: P(x)=λxe−λx!P(x) = \frac{\lambda^x e^{-\lambda}}{x!}P(x)=x!λxe−λ​

Continuous Probability Distributions

Properties of Continuous Probability Distributions

• The graph is a probability density function (pdf).

• Probability is represented by the area under the curve.

• The total area under the curve is always 1.

• The probability of a single value is zero.

Uniform Distribution

• Characteristics:

  • All values within an interval are equally likely.

  • Defined by minimum (aaa) and maximum (bbb) values.

• Formula:

  • Probability Density Function: f(x)=1b−af(x) = \frac{1}{b-a}f(x)=b−a1​

  • Mean: μ=a+b2\mu = \frac{a+b}{2}μ=2a+b​

  • Variance: σ2=(b−a)212\sigma^2 = \frac{(b-a)^2}{12}σ2=12(b−a)2​

Exponential Distribution

• Characteristics:

  • Measures the time between occurrences of an event.

  • Often used for reliability analysis (e.g., time until failure).

  • Memoryless property: The probability of an event occurring does not depend on past occurrences.

• Formula:

  • Probability Density Function: f(x)=λe−λxf(x) = \lambda e^{-\lambda x}f(x)=λe−λx

  • Mean: μ=1λ\mu = \frac{1}{\lambda}μ=λ1​

  • Variance: σ2=1λ2\sigma^2 = \frac{1}{\lambda^2}σ2=λ21​