Probability Distributions and Normal Distribution Applications in Statistics

Probability Distributions

Mean of a Probability Distribution

The mean (or expected value) of a probability distribution provides a measure of the central tendency of a random variable. It is calculated by multiplying each possible value of the random variable by its probability and summing the results.

• Formula: $\mu = \sum x \cdot P(x)$

• Example: For a batch of computers with probabilities for 0, 1, 2, 3, and 4 defective units, the mean is calculated as: $\mu = 0 \times 0.6561 + 1 \times 0.2916 + 2 \times 0.0486 + 3 \times 0.0036 + 4 \times 0.0001 = 0.4$

Standard Deviation of a Probability Distribution

The standard deviation measures the spread or variability of a probability distribution. It is the square root of the variance, which is calculated as the expected value of the squared deviation from the mean.

• Formula: $\sigma = \sqrt{\sum (x - \mu)^2 P(x)}$

• Example: For a given distribution, calculate each $(x - \mu)^2 P(x)$, sum, and take the square root. $\sigma = \sqrt{5.72 - 1.87^2} = 1.53$

Expected Value in Games of Chance

The expected value in games of chance is the average amount one can expect to win or lose per game in the long run. It is calculated by multiplying each outcome by its probability and summing the results.

• Example: If you pay $3 to roll a die and win $5 for rolling a 2 or 4, the expected value is:$E = 2 \times 5 \times \frac{1}{6} + 4 \times 0 = 1.67$ (rounded to the nearest cent)

Binomial Probability Distributions

Binomial Probability Formula

A binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials, each with probability p of success.

• Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Calculator Command: binompdf(n, p, k) for probability of exactly k successes. binomcdf(n, p, k) for cumulative probability up to k successes.

• Example: If 10% of people are left-handed, the probability that exactly 2 out of 7 are left-handed is: $P(X = 2) = \binom{7}{2} (0.1)^2 (0.9)^5 = 0.124$

Applications of Binomial Distribution

• Prime-time TV Example: Probability that fewer than three of seven adults watch prime-time TV live, with $p = 0.8$: $P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$ Use binomcdf(7, 0.8, 2) to compute cumulative probability.

• Test Passing Example: For a test of 10 true/false questions, probability of passing (at least 6 correct) with $p = 0.5$: $P(X \geq 6) = P(X = 6) + P(X = 7) + ... + P(X = 10)$ Use binomcdf(10, 0.5, 5) and subtract from 1.

Significantly High or Low Values

To determine if a value is significantly high or low in a binomial distribution, use the mean and standard deviation:

• Mean: $\mu = np$

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

• Significantly Low: $\mu - 2\sigma$

• Significantly High: $\mu + 2\sigma$

• Example: For $n = 1205$, $p = 0.098$: $\mu = 1205 \times 0.098 = 118.99$ $\sigma = \sqrt{1205 \times 0.098 \times 0.902} = 10.16$ Lower: $118.99 - 2 \times 10.16 = 98.67$ Upper: $118.99 + 2 \times 10.16 = 139.31$

Normal Distribution Applications

Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Z-scores measure how many standard deviations an observation is from the mean.

• Formula: $z = \frac{x - \mu}{\sigma}$

• Example: For a mean pregnancy length of 268 days and standard deviation of 15 days, the probability that a pregnancy lasts 272 days or longer: $z = \frac{272 - 268}{15} = 0.27$

Finding Probabilities Using the Normal Distribution

Probabilities for normal distributions are found using z-scores and standard normal tables or calculator functions.

• Formula: $P(a < X < b) = P(z_a < Z < z_b)$

• Example: For red blood cell counts with $\mu = 4.577$ million, $\sigma = 0.382$ million, probability that count is between 4.2 and 5.4 million: $z_1 = \frac{4.2 - 4.577}{0.382} = -0.99$ $z_2 = \frac{5.4 - 4.577}{0.382} = 2.16$ $P(-0.99 < Z < 2.16)$

Percentiles and Cutoff Values

Percentiles indicate the value below which a given percentage of observations fall. To find a cutoff value for a given percentile, use the inverse normal function.

• Example: Human body temperature is normally distributed with $\mu = 98.20$°F and $\sigma = 0.62$°F. To find the temperature separating the top 7% from the bottom 93%, find the z-score for the 93rd percentile and convert to temperature: $z_{0.93} = 1.475$ $x = \mu + z \sigma = 98.20 + 1.475 \times 0.62 = 99.12$°F

Sampling Distributions

When dealing with sample means, the standard deviation of the sampling distribution (standard error) is:

• Formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Example: For fish weights with $\mu = 16$ lb, $\sigma = 9$ lb, and $n = 19$, the probability that the mean weight is between 13.6 and 19.6 lb: $z_1 = \frac{13.6 - 16}{9/\sqrt{19}} = -0.8$ $z_2 = \frac{19.6 - 16}{9/\sqrt{19}} = 2$ $P(-0.8 < Z < 2) = 0.6731$

Summary Table: Key Probability Distributions

Distribution	Parameters	Mean	Standard Deviation	Example Application
Binomial	n, p	$np$	$\sqrt{np(1-p)}$	Number of successes in n trials
Normal	$\mu$, $\sigma$	$\mu$	$\sigma$	Heights, weights, test scores
Sampling Distribution of Mean	$\mu$, $\sigma$, n	$\mu$	$\frac{\sigma}{\sqrt{n}}$	Mean of sample means

Key Concepts and Calculator Functions

• binompdf(n, p, k): Probability of exactly k successes in n binomial trials.

• binomcdf(n, p, k): Cumulative probability of k or fewer successes.

• normalcdf(a, b, mu, sigma): Probability that a normal variable falls between a and b.

• invNorm(p, mu, sigma): Value corresponding to percentile p in a normal distribution.

Additional info:

• All probabilities should be rounded to three decimal places for binomial problems and four decimal places for normal problems, unless otherwise specified.

• Visual aids such as bell curves and shaded regions are commonly used to illustrate normal distribution problems.