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| -rw-r--r-- | houdre/hw5.tex | 8 |
1 files changed, 5 insertions, 3 deletions
diff --git a/houdre/hw5.tex b/houdre/hw5.tex index a952e76..54b8ed4 100644 --- a/houdre/hw5.tex +++ b/houdre/hw5.tex @@ -76,9 +76,11 @@ QED \q3 The density function $f(x)$ is proportional, so there is some constant -$c$ such that $$1 = f(x) = c\infint g(x)dx = 2c\int_1^\infty x^{-n}dx = -2c\left(-{x^{1-n}\over n-1}\right)\big|^\infty_1 = 2c\left({1\over -n-1}\right) \to c = {n-1\over2}.$$ +$c$ such that $$1 = \infint f(x)dx = c\infint g(x)dx = 2c\int_1^\infty +x^{-n}dx = 2c\left(-{x^{1-n}\over n-1}\right)\big|^\infty_1 = +2c\left({1\over n-1}\right) \to c = {n-1\over2}.$$ +$$f(x) = {n-1\over2}g(x),$$ +and this will look like a vertically stretched $1/x^2$ graph. The mean and variance of $X$ exist when $\E(X)$ and $\E(X^2)$ exist, respectively. The first exists when $n>2$ because for $n=2,$ |