Question

question 16 5.1 basics of probability distributions. compute the mean and standard deviation of a discrete random variable. 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.) x p(x) 0 0.74 1 0.14 2 0.09 3 0.02 4 0.01 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: message instructor post to forum

Explanation:

Step1: Calculate sum of probabilities

$0.74 + 0.14+0.09 + 0.02+0.01=1$

Step2: Calculate the mean ($\mu$)

The formula for the mean of a discrete - random variable is $\mu=\sum xP(x)$.
$\mu=(0\times0.74)+(1\times0.14)+(2\times0.09)+(3\times0.02)+(4\times0.01)$
$=0 + 0.14+0.18 + 0.06+0.04$
$=0.42$

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

The formula for the variance of a discrete - random variable is $\sigma^{2}=\sum(x - \mu)^{2}P(x)$.
$(0 - 0.42)^{2}\times0.74+(1 - 0.42)^{2}\times0.14+(2 - 0.42)^{2}\times0.09+(3 - 0.42)^{2}\times0.02+(4 - 0.42)^{2}\times0.01$
$=( - 0.42)^{2}\times0.74+(0.58)^{2}\times0.14+(1.58)^{2}\times0.09+(2.58)^{2}\times0.02+(3.58)^{2}\times0.01$
$=0.1764\times0.74 + 0.3364\times0.14+2.4964\times0.09+6.6564\times0.02+12.8164\times0.01$
$=0.130536+0.047096+0.224676+0.133128+0.128164$
$=0.6636$

Step4: Calculate the standard deviation ($\sigma$)

$\sigma=\sqrt{\sigma^{2}}=\sqrt{0.6636}\approx0.81$

Answer:

Sum of probabilities: $1$
Mean: $0.42$
Standard deviation: $0.81$