Tiêu đề 32 Common Continuous Probability Distributions.pdf ✅

COMMON CONTINUOUS PROBABILITY DISTRIBUTIONS 1. Continuous Uniform Distribution The probability distribution is rectangular in its graph since the probability values are always the same for any point in the distribution. The function is given by f(x; A,B) = { 1 B − A , A ≤ x ≤ B 0, elsewhere The mean of the distribution: μ = A + B 2 The variance of the distribution: σ 2 = (B − A) 2 12 2. Normal Distribution Also known as the Gaussian distribution, it is a distribution having a bell-shaped curve that depends on two parameters: the mean and the standard deviation. 2.1. General Normal Distribution The density of the normal random variable X with mean μ and variance σ 2 is n(x; μ, σ) = 1 σ√2π e − 1 2 ( x−μ σ ) 2 Defined in the interval −∞ < x < ∞. 2.2. Standard Normal Distribution Normal curves are transformed to the standard normal distribution by standardization to simplify the computation of probabilities. z = x − μ σ This formula changes the distribution to the parameters: μ = 0, σ = 1. Hence, the probabilities are as follows: P(x1 < x < x2 ) = P(z1 < z < z2) 2.3. Use a Calculator to find Areas under the Normal Curve To determine the normal curve probabilities, use the distribution function in the statistics mode of a calculator. 2.3.1. Setting up the function
Enter “stat” mode Do not pick any of the functions below. Press CA/AC Enter the statistics functions window (using APPS or Shift-Stat) Choose “Distr”, which is the distribution mode. 2.3.2. The P, Q, and R functions • P function – The probability P(z < z1) • Q function – The probability P(0 < z < z1 ) • R function – The probability P(z > z1 ) - Examples: (Note that the standard normal curve follows Probability Graph Calculator P(z < −0.8)
P(0 < z < 2.1) P(z > −1.7) P(0.5 < z < 1.6) P(−1 < z < 1)
2.3.3. The t Function The → t function in the distribution mode is helpful for general normal distributions. The standardization formula may still be beneficial, but this function may eliminate its use. - Example: Find the probability P(x > 60) for a normal distribution with a mean of 56 and a standard deviation of 8. [SOLUTION] Enter stat mode with a one-variable function. To ensure that the distribution results in a mean of 56 and a standard deviation of 8, two entries must be: μ + σ, μ − σ, i.e., 56 − 8 = 48, 56 + 8 = 64. Press CA/AC and proceed to distribution mode Since P(x > 60) uses the R function, the syntax must be Here, 60 → t is the standardization function for x = 60. To verify, z = 60 − 56 8 = 0.5