Question

Math

Posted 6 months ago

```
In one online store, $12 \%$ of the orders are placed by new customers. Let $N$ be the number of orders placed in a day to reach the store's first new customer. Assume each order's customer status is independent.
Find the mean and standard deviation of $N$.
Round your answers to one decimal place.
Mean: $\square$ orders
Standard deviation: $\square$ orders
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

To find the mean of the geometric distribution with probability $p = 0.12$, we use the formula $\mu = \frac{1}{p}$

step 2

Substituting $p = 0.12$ into the formula gives us $\mu = \frac{1}{0.12}$

step 3

Calculating the mean, we get $\mu = \frac{1}{0.12} = 8.33333$

step 4

To find the standard deviation of the geometric distribution, we use the formula $\sigma = \frac{\sqrt{1-p}}{p}$

step 5

Substituting $p = 0.12$ into the formula gives us $\sigma = \frac{\sqrt{1-0.12}}{0.12}$

step 6

Calculating the standard deviation, we get $\sigma = \frac{\sqrt{0.88}}{0.12} = 7.81736$

step 7

Round the mean and standard deviation to one decimal place: Mean = 8.3 orders, Standard deviation = 7.8 orders

1 Answer

Mean: 8.3 orders

Standard deviation: 7.8 orders

Key Concept

Geometric Distribution

Explanation

The mean of a geometric distribution with probability $p$ is given by $\frac{1}{p}$, and the standard deviation is given by $\frac{\sqrt{1-p}}{p}$. These formulas are used to calculate the expected number of trials until the first success and the variability of this number, respectively.

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