# Probability density functions cumulative df and

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Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Normal Random Variable Normal Table Percentile Critical Values Percentile of an Arbitrary Normal Distribution Normal Approximation for Binomial Distribution Outline Normal Distribution Normal Random Variable Normal Table Percentile Critical Values Percentile of an Arbitrary Normal Distribution Normal Approximation for Binomial Distribution Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Normal Random Variable Normal Table Percentile Critical Values Percentile of an Arbitrary Normal Distribution Normal Approximation for Binomial Distribution Approximation for Binomial Distribution Let X be a binomial rv based on n trials with success probability p . Then if the binomial probability histogram is not too skewed, X has approximately a normal distribution with μ = np and σ = npq . In particular, for x = a positive value of X P ( X x ) = B ( x ; n , p ) area under the normal curve to the left of x+.5 = Φ x + . 5 - np npq In practice, the approximation is adequate provided that both np 10 and nq 10, since there is then enough symmetry in the underlying binomial distribution. Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Normal Random Variable Normal Table Percentile Critical Values Percentile of an Arbitrary Normal Distribution Normal Approximation for Binomial Distribution Example 4.20 Suppose that 25% of all students at a large public university receive financial aid. Let X be the number of students in a random sample of size 50 who receive financial aid, so that p = . 25. Then μ = 12 . 5 and σ = 3 . 06 . Since np = (50( . 25) = 12 . 5 > 10 and nq = 37 . 5 10, the approximation can safely be applied. The probability that at most 10 students receive aid is P ( X 10) = B (10; 50 , . 25) = Φ ( 10 + . 5 - 12 . 5 3 . 06 ) = Φ ( . 65) = . 2578 Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Normal Random Variable Normal Table Percentile Critical Values Percentile of an Arbitrary Normal Distribution Normal Approximation for Binomial Distribution Example4.20 (cont) P ( 4 < X 15) Z 15 . 5 4 . 5 φ ( x ; 15 . 5 , 3 . 06) dx Φ ( 15 . 5 - 12 . 5 3 . 06 ) - Φ ( 4 . 5 - 12 . 5 3 . 06 ) = . 8320 P ( X = 10 ) Z 10 . 5 9 . 5 φ ( x ; 15 . 5 , 3 . 06) dx = Φ ( 10 . 5 - 12 . 5 3 . 06 ) - Φ ( 9 . 5 - 12 . 5 3 . 06 ) = Chapter 4: Continuous RV and Probability Distributions
Probability Density Functions Cumulative DF and Expected Values Normal Distribution Exponential Distribution Gamma Distribution Chi-Square distribution Probability Plot Normal Random Variable Normal Table Percentile Critical Values