Probability, Normal Distributions, and Discrete Random Variables: Study Notes

Probability and Normal Distributions

Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. To compare values from different normal distributions, we convert them to z-scores, which measure how many standard deviations a value is from the mean.

• Z-score formula:

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

• For sample means (Central Limit Theorem):

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

• Finding probabilities: Use z-tables to find the probability that a value is above or below a certain z-score.

• Example: If systolic blood pressure for women aged 18-24 is normally distributed with $\mu = 114.8$ and $\sigma = 13.1$, the probability that a randomly selected woman has a systolic blood pressure greater than 120 is found by:

$z = \frac{120 - 114.8}{13.1} = 0.40$

$P(X > 120) = P(z > 0.40) = 1 - P(z < 0.40) = 1 - 0.6554 = 0.3446$

• For sample means: If 40 women are sampled, the standard error is $\sigma/\sqrt{n}$.

$z = \frac{120 - 114.8}{13.1/\sqrt{40}} = 2.51$

$P(\bar{X} > 120) = 1 - P(z < 2.51) = 1 - 0.9939 = 0.0061$

Percentiles and Inverse Normal Calculations

To find the value corresponding to a given percentile in a normal distribution, use the z-score for that percentile and solve for $X$:

$X = z\sigma + \mu$

• Example: For the 90th percentile ($z = 1.28$), $\mu = 560$, $\sigma = 85$:

$X = 1.28 \times 85 + 560 = 668.8$

Probability with Contingency Tables

Joint, Marginal, and Conditional Probability

Contingency tables summarize data for two categorical variables. Probabilities can be calculated as follows:

• Joint probability: Probability of two events both occurring (e.g., being male and a smoker).

• Marginal probability: Probability of a single event (e.g., being male).

• Conditional probability: Probability of one event given another (e.g., probability of being a non-smoker given male).

Example Table:

	Female	Male	Total
Smoker	38	51	89
Non-Smoker	127	10	137
Total	165	61	226

• Probability of selecting a male smoker: $P(\text{male smoker}) = \frac{51}{226} = 0.2257$

• Probability of not a smoker or male: $P(\text{not a smoker or male}) = \frac{204 + 49 + 84}{226}$ (Additional info: This sums non-smokers and all males)

Probability without Replacement

Dependent Events in Sampling

When sampling without replacement, the probability of subsequent events changes because the sample space is reduced.

• Example: Probability of drawing two face cards in a row from a standard deck:

$P(\text{2 face cards}) = \frac{12}{52} \times \frac{11}{51} = 0.0498$

Discrete Probability Distributions

Mean (Expected Value) of a Discrete Random Variable

The mean or expected value of a discrete random variable $X$ with probability distribution $P(x)$ is:

$\mu = E(X) = \sum xP(x)$

• Example: For the following distribution:

x	P(x)
0	0.40
1	0.31
2	0.23
4	0.02
5	0.04

$E(X) = 0 \times 0.40 + 1 \times 0.31 + 2 \times 0.23 + 4 \times 0.02 + 5 \times 0.04 = 0 + 0.31 + 0.46 + 0.08 + 0.20 = 1.05$

Additional info: The sum in the image is 0.97, but the correct calculation is shown above.

Binomial Probability

Binomial Probability Formula

The binomial probability gives the probability of exactly $x$ successes in $n$ independent Bernoulli trials with probability $p$ of success:

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

• Example: For $n = 10$, $p = 0.6$, $x = 8$:

$P(8) = \binom{10}{8} (0.6)^8 (0.4)^2 = 45 \times (0.6)^8 \times (0.4)^2 = 0.1209$