The mgf of a random variable X is ((2)/(3)+(1)/(3) e^t)^9. (a) Show that P(μ-2 σ<X<μ+2 σ)=∑x=1^5

step 1 Okay, in this question we have to show probability of u minus 2 svema in between x and u plus 2

The mgf of a random variable X is ((2)/(3)+(1)/(3) e^t)^9....

The mgf of a random variable $X$ is $\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9}$. (a) Show that $$ P(\mu-2 \sigma<X<\mu+2 \sigma)=\sum_{x=1}^{5}\left(\begin{array}{l} 9 \\ x \end{array}\right)\left(\frac{1}{3}\right)^{x}\left(\frac{2}{3}\right)^{9-x} $$ (b) Use $\mathrm{R}$ to compute the probability in Part (a).

The mgf of a random variable $X$ is $\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9}$.
(a) Show that
$$
P(\mu-2 \sigma<X<\mu+2 \sigma)=\sum_{x=1}^{5}\left(\begin{array}{l}
9 \\
x
\end{array}\right)\left(\frac{1}{3}\right)^{x}\left(\frac{2}{3}\right)^{9-x}
$$
(b) Use $\mathrm{R}$ to compute the probability in Part (a).

Key Concepts

Probability Computation using Statistical Software

Statistical software, such as R, provides built-in functions to compute probabilities for common distributions. For a binomial distribution, functions like dbinom and pbinom can be used to sum probabilities over specific ranges, thereby facilitating the calculation of the desired probability interval.

Moment Generating Function (mgf)

The mgf uniquely characterizes a probability distribution and can be used to derive the moments (such as the mean and variance) of the distribution by differentiating the mgf. In the context of identifying a distribution, recognizing the form of the mgf helps determine which standard distribution is applicable.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Recognizing the mgf as the one corresponding to the binomial form allows for applying known formulas and properties associated with the binomial model.

Mean and Standard Deviation for the Binomial Distribution

For a binomial random variable, the mean is given by np and the variance by npq (where q = 1-p). These parameters are essential to assess the spread of the distribution and, in this case, to determine the interval around the mean (expressed as ? ± 2?) within which the probability is calculated.

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