SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the chi-square distribution with parameters v FIGuRE 5. Ch df and cdf Probability Density Function Cumulative Distribution Function 0 0 0.02 X 2.2. Properties of the chi-square random variable 2.2.1. x and N(0, 1). Consider n independent random variables IfX;~N(0,1) (12) It can also be shown that IfX;~N(0,1) then >(Xi-X)N x(n-1) 13) because this is the sum of (n-1)independent random variables given that X and(n-1)of the x's 222.x2andN(1,a2) IfX1~N(μ,a2) 1.2 th (14) and x-12 ~x2(n-1) 2.2.3. Sums of chi-square random variables. If yi and y2 are independently distributed as x(v1) (1+D2)6 SOME SPECIFIC PROBABILITY DISTRIBUTIONS The following diagram shows the pdf and cdf for the chi-square distribution with parameters ν =10. Figure 5. Chi-square pdf and cdf 0 10 20 30 0 0.02 0.04 0.06 0.08 0.1 Probability Density Function X f(X) 0 10 20 30 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) 2.2. Properties of the chi-square random variable. 2.2.1. χ2 and N(0,1). Consider n independent random variables. If Xi ∼ N (0, 1) i = 1, 2, ... , n then Pn i=1 X2 i ∼ χ2(n) (12) It can also be shown that If Xi ∼ N (0, 1) i = 1, 2, ... , n then Pn i=1 (Xi − X¯) 2 ∼ χ2(n − 1) (13) because this is the sum of (n-1) independent random variables given that X¯ and (n-1) of the x’s are independent. 2.2.2. χ2 and N(µ,σ2). If Xi ∼ N (µ, σ2) i = 1, 2, ... , n then Xn i=1 Xi − µ σ 2 ∼ χ2(n) (14) and Xn i=1 Xi − X¯ σ 2 ∼ χ2 (n − 1) 2.2.3. Sums of chi-square random variables. If y1 and y2 are independently distributed as χ2(ν1) and χ2(ν2), respectively, then y1 + y2 ∼ χ2(ν1 + ν2). (15)