Question

The mgf of a random variable \(X\) is \(\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9}\). (a) Show that $$ P(\mu-2 \sigma

Short answer

To solve this problem, first, the mean \( \mu \) and standard deviation \( \sigma \) of the random variable \( X \) are calculated from its mgf. Then the given probability function is derived using the formula of a binomial distribution. Finally, the sum of the probabilities for \( x = 1 \) to 5 is calculated using R. The code necessary to do this is: `prob <- sum(dbinom(1:5, size=9, prob=1/3))`.

Step-by-step solution

Calculate Mean and Standard Deviation

The moment generating function is given by \(M(t) = \left(\frac{2}{3}+\frac{1}{3}e^{t}\right)^{9}\). The mean or expectation \( \mu \) of a random variable \( X \) is given by the first derivative of the MGF evaluated at t = 0, i.e., \( \mu = M'(0) \). The variance \( \sigma^{2} \) of a random variable \( X \) is given by the second derivative of the MGF evaluated at t = 0, i.e., \( \sigma^{2} = M''(0) - [M'(0)]^2 \). The standard deviation \( \sigma \) is the square root of the variance, i.e., \( \sigma = \sqrt{\sigma^{2}} \).

Derive the Probability Function

The given function in the problem is that of a binomial distribution, that we know by the formula \[P (X = x) = \binom{n}{x} p^{x} (1 - p)^{n - x}\]. Here, the number of trials \(n = 9\), success probability \(p = \frac{1}{3}\), and the failure probability will be \(q = 1 - p = \frac{2}{3}\). The summation is running from \(x = 1\) to 5, which suggests that we sum up the probabilities of \( x \) being 1, 2, ..., and 5. Now we substitute these values into the binomial distribution sum to derive the required probability function.

Calculate The Probability Using R

Using the derived function, calculate the probability. The R code necessary to calculate this probability would be: \n\n`prob <- sum(dbinom(1:5, size=9, prob=1/3))`\n\nThis code calculates the binomial probabilities for \( x = 1 \) to 5 with parameters \( n =9 \) and \( p = \frac{1}{3} \), and sums them up. Execute the code in R to get the desired probability.