p>=100
mean <- 0
var <- 1/4
sd <- sqrt(var)
n <- 364
Y1 <- 100
Y365 <- 100
x <- (Y365- Y1) /sqrt(n)
p_norm <- pnorm(x, mean, sd, lower.tail = FALSE)
paste0('The probability is ' , round(p_norm, 4))## [1] "The probability is 0.5"p>=110
mean <- 0
var <- 1/4
sd <- sqrt(var)
n <- 364
Y1 <- 100
Y365 <- 110
x <- (Y365- Y1) /sqrt(n)
p_norm <- pnorm(x, mean, sd, lower.tail = FALSE)
paste0('The probability is ' , round(p_norm, 4))## [1] "The probability is 0.1473"p>=120
mean <- 0
var <- 1/4
sd <- sqrt(var)
n <- 364
Y1 <- 100
Y365 <- 120
x <- (Y365- Y1) /sqrt(n)
p_norm <- pnorm(x, mean, sd, lower.tail = FALSE)
paste0('The probability is ' , round(p_norm, 4))## [1] "The probability is 0.018"Calculate the expected value and variance of the binomial distribution using the moment generating function.
The binomial is defined as; Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is p, then the binomial probability is: b(x,n,p)
\[\begin{equation} \binom{n}{x} p^xq^{n−x} \end{equation}\]
where q=(1−p)
so the moment generating function is:
\[M(t)=\sum_{x=0}^n e^{tx} \binom{n}{x} p^xq^{n−x}= \sum_{x=0}^n \binom{n}{x} (pe^t)^xq^{n−x}=(pe^t+q)^n \]
\[M'(t)=n(pe^t+q)^n−pe^t \] \[E(X)=M'(0)=np \]
\[M''(t)=n[1−p+pe^t]^{n−1}(pet)+(pe^t)n(n−1)[1−p+pe^t]^{n−2}(pe^t) \]
\[E(X^2)=M''(0)=n(n−1)p^2+np \]
\[Var(X)=E(X^2)−E(X)^2 = n(n−1)p^2+np−(np)^2 \]
\[=(n^2p^2−1np^2)+np−(np)^2 \]
\[=(np)^2−np^2+np−(np)^2 \]
\[=np−np^2 = np(1−p) \]
\[=npq \]
Calculate the expected value and variance of the exponential distribution using the moment generating function.
The exponential PDF is defined as
\[λe^{−xλ} \]
so the moment generating function is:
\[M(t)=∫_0^\infty e^{tx}λe^{−xλ}dx \]
\[=λ∫_0^\infty e^{−x(λ−t)} \]
\[ =−λ \frac {e^{−x(λ−t)} }{ (λ−t) } |_0^\infty \]
\[=\frac{λ}{λ−t}\]
\[M'(t)=\frac{λ}{(λ−t)} 2E(X)\]
\[=M'(0)=\frac{λ}{λ^2}\]
\[=\frac{1}{λ}\]
\[M''(t)=\frac{2λ}{(λ−t)^3}\]
\[E(X^2)=M''(0)=\frac{2λ}{λ^3}=\frac{2}{λ^2}\]
\[Var(X)=E(X^2)−E(X)^2\]
\[=\frac{2}{λ^2}−\frac{1}{λ^2}=\frac{1}{λ^2}\]