this table gives approximate probabilities for scoring 0, 1, 2, 3, or 4 runs in one inning of major league baseball. (although it is possible to score more than 4 runs in one inning, the probability is very small, so it is ignored in this question.)
round all answers to 2 decimal places.
compute the sum of these probabilities:
compute the mean number of runs/inning, using this probability distribution:
compute the standard deviation of this probability distribution:
question help: written example message instructor submit question

Question

Accepted Answer

# Explanation:
## Step1: Compute the sum of probabilities
The sum of probabilities in a probability distribution must be $1$. We check if $0.7+0.14 + 0.09+0.04+0.03=1$.
$$0.7+0.14+0.09+0.04+0.03=(0.7 + 0.14)+(0.09+0.04 + 0.03)=0.84+(0.16)=1$$

## Step2: Compute the mean $\mu$
The formula for the mean of a discrete probability distribution is $\mu=\sum_{x}xP(x)$.
$$
\begin{align*}
\mu&=(0\times0.7)+(1\times0.14)+(2\times0.09)+(3\times0.04)+(4\times0.03)\
&=0 + 0.14+0.18+0.12 + 0.12\
&=0.56
\end{align*}
$$

## Step3: Compute the variance $\sigma^{2}$
The formula for the variance is $\sigma^{2}=\sum_{x}(x-\mu)^{2}P(x)$
$$
\begin{align*}
(0 - 0.56)^{2}\times0.7&=( - 0.56)^{2}\times0.7=0.3136\times0.7 = 0.21952\
(1-0.56)^{2}\times0.14&=(0.44)^{2}\times0.14=0.1936\times0.14=0.027104\
(2 - 0.56)^{2}\times0.09&=(1.44)^{2}\times0.09=2.0736\times0.09=0.186624\
(3-0.56)^{2}\times0.04&=(2.44)^{2}\times0.04=5.9536\times0.04=0.238144\
(4 - 0.56)^{2}\times0.03&=(3.44)^{2}\times0.03=11.8336\times0.03=0.355008
\end{align*}
$$
$$
\begin{align*}
\sigma^{2}&=0.21952+0.027104 + 0.186624+0.238144+0.355008\
&=(0.21952+0.027104)+(0.186624+0.238144)+0.355008\
&=0.246624+0.424768+0.355008\
&=1.0264
\end{align*}
$$

## Step4: Compute the standard deviation $\sigma$
The standard deviation $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2} = 1.0264$, then $\sigma=\sqrt{1.0264}\approx1.01$

# Answer:
The sum of probabilities is $1$, the mean is $0.56$, and the standard deviation is $1.01$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Compute the sum of probabilities

Step2: Compute the mean \(\mu\)

Step3: Compute the variance \(\sigma^{2}\)

Step4: Compute the standard deviation \(\sigma\)

Answer: