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7.1 continuous random variables (probability will be area under a curve) __________________________________________________________________ Uniform Distribution EX: You are expecting a phone call during the next hour. The call could come at any time (x) during the hour, with any time interval being equally likely. Thus, 0 ≤ x ≤ 60. Note: this is a uniform probability distribution (intervals of the same length are equally likely). Graph this distribution. Note the curve, y = 1/60, is the probability density function (pdf). __________________________________________________________________ Probability density function (pdf) 1. Area under the curve = 1. 2. f(x) ≥ 0 for all x. The area under the graph of a pdf over an interval represents the probability of observing a value of the random variable in that interval. Normal Distribution (bell-shaped) Properties of a Normal Distribution 1. 2. 3. 4. 5. Symmetric about the mean. (mean = median) Inflection points at μ-σ and μ+σ. Area under the curve is 1. (Areas to the left and right of μ are each ½) Horizontal asymptotes – horizontal axis. Empirical Rule (68%, 95%, 99.7%). Graph: __________________________________________________________________ Create/compare graphs: a) μ = 0, σ = 3 (note actual graphs would have different heights, spread, etc.) b) μ = 5, σ = 0.2 c) μ = 100, σ = 15 (interpret a score of 115, proportion or probability) __________________________________________________________________ Example: Assume IQ scores have a normal distribution with μ = 100 and σ = 15. Use the Empirical Rule to find the following probabilities. a) Find P(X > 100) compare to P(X ≥ 100) b) Find P(85 < X < 115) c) Find P(X > 130) d) Find P(X < 85) Use the graph: __________________________________________________________________ Standardizing a Normal Random Variable Suppose X is normally distributed with mean, μ, and standard deviation, σ. Z= 𝑋−𝜇 𝜎 this variable is normally distributed with μ = 0 and σ = 1. __________________________________________________________________ Example: 1st: Find z-scores for IQ’s of 115, 130, 85, 106, 140. 2nd: a) Find P(Z > 0) b) Find P(-1 < Z < 1) c) Find P(Z > 2) d) Find P(Z < -1) Repeat using normalcdf on calculator (note use of 1E99) Also find: e) P(-0.4 < Z < 1.2) f) P(Z < 0.46) g) P(Z > -2.15) __________________________________________________________________ Note < vs. ≤ for continuous distributions. __________________________________________________________________