Unit 4: Probability Distributions

Random Variables, Expectation, Binomial, Poisson & Normal Distributions

Random Variable ($$X$$)	A real-valued function whose domain is the sample space of a random experiment.
Types	1. Discrete: Takes finite or countably infinite values (e.g., number of heads). 2. Continuous: Takes any value within an interval (e.g., height, weight).
Probability Distribution	A system of numbers $$X$$(values) and$$P(X)$$ (probabilities) such that: 1. $$P(x_i) \ge 0$$ 2. $$\sum P(x_i) = 1$$

Expected Value ($$E(X)$$or$$\mu$$)	The weighted average (mean) of the random variable. $$E(X) = \sum_{i=1}^{n} x_i p_i$$
Variance ($$Var(X)$$or$$\sigma^2$$)	$$Var(X) = E(X^2) – [E(X)]^2$$ Where $$E(X^2) = \sum x_i^2 p_i$$
Standard Deviation ($$\sigma$$)	$$SD(X) = \sqrt{Var(X)}$$

Used for Bernoulli Trials (finite trials, independent, only two outcomes: Success/Failure).

Parameters	$$n$$ = Number of trials $$p$$ = Probability of success $$q$$ = Probability of failure ($$q = 1 – p$$)
Probability Formula	$$P(X = r) = {}^nC_r p^r q^{n-r}$$
Key Statistics	Mean: $$\mu = np$$ Variance: $$\sigma^2 = npq$$ Standard Deviation: $$\sigma = \sqrt{npq}$$

Used for rare events where $$n$$is very large and$$p$$ is very small ($$np$$ is finite).

Parameter

$$\lambda$$ (Lambda) = Average rate of occurrence ($$\lambda = np$$)

Probability Formula

$$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$$

(where $$e \approx 2.71828$$)

Key Statistics

Mean: $$\lambda$$
Variance: $$\lambda$$
(In Poisson, Mean = Variance)

Continuous probability distribution (Bell Curve).

Characteristics

1. Bell-shaped and symmetric about the Mean.
2. Mean = Median = Mode.
3. Total area under the curve = 1.

Standard Normal Variate ($$Z$$)

$$Z = \frac{X – \mu}{\sigma}$$

($$X$$: Value, $$\mu$$: Mean, $$\sigma$$: S.D.)

Standard Normal Distribution

When $$\mu = 0$$and$$\sigma = 1$$.