Discrete and Normal Probability Distributions: Study Notes

Discrete Probability Distributions

Probability Distribution of a Discrete Random Variable

A discrete probability distribution lists each possible value a discrete random variable can take, along with its probability. The sum of all probabilities must equal 1.

• Random Variable (X): A variable whose value is a numerical outcome of a random phenomenon.

• Probability of X: $P(X = x)$ for each possible value $x$.

• Requirements:

  • Each probability $P(X = x)$ must be between 0 and 1.

  • The sum of all probabilities must be 1: $\sum P(X = x) = 1$.

• Example Table:

X	P(X)
0	0.2
1	0.5
2	0.3

Additional info: Table values are illustrative; actual values may differ in specific problems.

Mean (Expected Value) and Standard Deviation

• Mean (Expected Value): $\mu_X = \sum [x \cdot P(X = x)]$

• Standard Deviation: $\sigma_X = \sqrt{\sum [(x - \mu_X)^2 \cdot P(X = x)]}$

• Interpretation: The mean gives the long-run average value of the random variable; the standard deviation measures the spread.

• Example: For the table above, $\mu_X = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$

Binomial Probability Distribution

Characteristics of a Binomial Experiment

A binomial experiment is a statistical experiment with the following properties:

• Fixed number of trials ($n$)

• Each trial has two possible outcomes: success or failure

• Probability of success ($p$) is constant for each trial

• Trials are independent

Binomial Probability Formula

• Probability of exactly $x$ successes in $n$ trials:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

• $\binom{n}{x}$ is the binomial coefficient: $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

• Example: $n = 5$, $p = 0.3$, $x = 2$

$P(X = 2) = \binom{5}{2} (0.3)^2 (0.7)^3 = 10 \times 0.09 \times 0.343 = 0.3087$

Mean and Standard Deviation of Binomial Distribution

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{np(1-p)}$

• Example: For $n = 10$, $p = 0.4$:

$\mu = 10 \times 0.4 = 4$ $\sigma = \sqrt{10 \times 0.4 \times 0.6} = \sqrt{2.4} \approx 1.55$

Normal Probability Distribution

Properties of the Normal Distribution

• Bell-shaped and symmetric about the mean $\mu$

• Mean, median, and mode are all equal

• Empirical Rule:

  • About 68% of data within 1 standard deviation ($\sigma$) of the mean

  • About 95% within 2$\sigma$

  • About 99.7% within 3$\sigma$

Standard Normal Distribution and Z-Scores

• Standard Normal Distribution: A normal distribution with $\mu = 0$ and $\sigma = 1$

• Z-score: The number of standard deviations a value $x$ is from the mean

$z = \frac{x - \mu}{\sigma}$

• Finding Probabilities: Use Z-tables to find the area under the curve to the left of a given z-score

• Example: If $x = 85$, $\mu = 80$, $\sigma = 5$:

$z = \frac{85 - 80}{5} = 1$

Look up $z = 1.00$ in the Z-table to find $P(Z < 1.00) \approx 0.8413$

Applications of the Normal Distribution

• Finding probabilities for intervals (e.g., $P(a < X < b)$)

• Finding percentiles and cut-off values

• Solving real-world problems involving normally distributed variables

Sampling Distributions

Sampling Distribution of the Sample Mean

• Definition: The probability distribution of all possible sample means of a given size from a population

• Central Limit Theorem (CLT): For large $n$, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution

• Mean of Sampling Distribution: $\mu_{\bar{x}} = \mu$

• Standard Deviation (Standard Error): $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Example: If $\mu = 50$, $\sigma = 10$, $n = 25$:

$\mu_{\bar{x}} = 50$ $\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2$

Using the Normal Approximation

• For large $n$, use the normal distribution to approximate probabilities for sample means

• Convert sample mean to z-score:

$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

• Find probabilities using the standard normal table

Summary Table: Key Formulas

Concept	Formula
Mean of Discrete RV	$\mu_X = \sum [x \cdot P(X = x)]$
SD of Discrete RV	$\sigma_X = \sqrt{\sum [(x - \mu_X)^2 \cdot P(X = x)]}$
Binomial Probability	$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$
Mean of Binomial	$\mu = np$
SD of Binomial	$\sigma = \sqrt{np(1-p)}$
Z-score	$z = \frac{x - \mu}{\sigma}$
Mean of Sample Mean	$\mu_{\bar{x}} = \mu$
SD of Sample Mean	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Additional info: These notes synthesize and expand upon the handwritten content for clarity and completeness.