# Expected Value and Variance PDF fx(x) = 3/x^4, x > 1

`````` #define a vector
> X <- 0:10
>
>
> #define the probability
> PDF <- dbinom(x, 10, .408)
# I calculated the formula give in instruction #2 fx(x)= 3/x^4,x > 1 ( is this correct
>
> #adding the sum code for the PDF
> sum(PDF)
 1
>
>
> #not sure if this is the proper format of inputting the PDF given in the HW file
> f <- function(x) 3/ ((x^2), (x > 1))
>g <- function(x) x*f(x)
>h <- function(x) x^4 * f(x)
> #it gives an error when I run it.
>
>
> #compute E(x)
> EX <- sum(x*PDF);
> EX
 4.08
>
> #computing the variance
> varX <- sum((X-EX)^2*PDF)
>
> #computing
> EX <- integrate(h,
+                  lower = 1,
+                  upper = Inf)\$value - (EX)^2

# I have to calculate the expected value and variance of the fx(x) = 3/x^4, x > 1
``````

# Expected value and variance of f(x) = 3/x^4, 1 < x < ∞

E(x) = ∫xf(x)dx = ∫x3*(x^-4)dx = ∫3*(x^-3)dx = 3*(x^-2)/(-2) = -1.5*(x^-2) = 1.5
E(x^2) = ∫(x^2)f(x)dx = ∫(x^2)3(x^-4)dx = ∫3(x^-2)dx = 3*(x^-1)/(-1) = -3*(x^-1) = 3
Var(x) = E(x^2) - (E(x))^2 = 3 - 1.5^2 = .75

# E(x)

integrand <- function(x) {x3(x^-4)} # note that the two asterisks don't show up for some reason
Ex <- integrate(integrand, lower = 1, upper = Inf)
Ex\$value

# E(x^2)

integrand <- function(x) {(x^2)3(x^-4)} # note that the two asterisks don't show up for some reason
Ex2 <- integrate(integrand, lower = 1, upper = Inf)
Ex2\$value

Var = Ex2\$value - (Ex\$value)^2
Var

@fcas80, I think the intention in the homework problem is that the pdf is binomial and the f(x) functions are transformations. (But I could be wrong.) Also not clear to me, but maybe the support is 0 to 10, not to infinity.

Startz, good point. Let's hear from the original poster.

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