name:
date:
directions: decide whether the information represents a discrete random variable or a continuous random variable. explain your reasoning.
1. ( x ) represents the number of motorcycle accidents in one year in california
2. ( x ) represents the length of time it takes to get to work
3. ( x ) represents the volume of blood drawn for a blood test
4. ( x ) represents the number of rainy days in the month of july in orlando, florida
5. ( x ) represents the number of home theater systems sold per month at an electronics store
6. ( x ) represents the tension at which a randomly selected guitar’s strings have been strung
7. ( x ) represents the amount of snow (in inches) that fell in nome, alaska last winter
8. ( x ) represents the total number of die rolls required for an individual to roll a five
directions: use the frequency distribution to construct a probability distribution. then, find the mean and standard deviation of the probability distribution.
9. dogs the number of dogs per household in a small town
| dogs | 0 | 1 | 2 | 3 | 4 | 5 |
| household | 1491 | 425 | 168 | 48 | 29 | 14 |

Question

Accepted Answer

### Problem 1:
# Brief Explanations:
A discrete random variable is one that can take on a countable number of distinct values (like 0, 1, 2, ...). The number of motorcycle accidents in a year can only be whole numbers (you can't have a fraction of an accident), so it's countable.
# Answer:
Discrete random variable. Reason: The number of motorcycle accidents is a countable value (0, 1, 2, ...).

### Problem 2:
# Brief Explanations:
A continuous random variable can take on any value within an interval (e.g., 5.2 minutes, 10.75 minutes). The length of time to get to work can be any real number (not just whole numbers), like 15.5 minutes or 20.25 minutes, so it's continuous.
# Answer:
Continuous random variable. Reason: Time length can take any real - valued measurement within an interval.

### Problem 3:
# Brief Explanations:
The volume of blood drawn can be any real number within a certain range (e.g., 5.2 milliliters, 10.75 milliliters). It's not restricted to whole numbers, so it's a continuous random variable.
# Answer:
Continuous random variable. Reason: Blood volume can take any real - valued measurement within a range.

### Problem 4:
# Brief Explanations:
The number of rainy days in a month is a countable value (0, 1, 2, ..., up to the number of days in July). You can't have a fraction of a rainy day, so it's a discrete random variable.
# Answer:
Discrete random variable. Reason: The number of rainy days is a countable value (0, 1, 2, ...).

### Problem 5:
# Brief Explanations:
The number of home theater systems sold per month is a countable value (0, 1, 2, ...). You can't sell a fraction of a system, so it's a discrete random variable.
# Answer:
Discrete random variable. Reason: The number of systems sold is a countable value (0, 1, 2, ...).

### Problem 6:
# Brief Explanations:
The tension at which a guitar's strings are strung can be any real number within a certain range (e.g., 5.2 Newtons, 10.75 Newtons). It's not restricted to whole numbers, so it's a continuous random variable.
# Answer:
Continuous random variable. Reason: Tension can take any real - valued measurement within a range.

### Problem 7:
# Brief Explanations:
The amount of snow that fell can be any real number within a certain range (e.g., 5.2 inches, 10.75 inches). It's not restricted to whole numbers, so it's a continuous random variable.
# Answer:
Continuous random variable. Reason: Snow amount can take any real - valued measurement within a range.

### Problem 8:
# Brief Explanations:
The number of die rolls required to roll a five is a countable value (1, 2, 3, ...). You can't have a fraction of a die roll, so it's a discrete random variable.
# Answer:
Discrete random variable. Reason: The number of die rolls is a countable value (1, 2, 3, ...).

### Problem 9:
### Step 1: Calculate the total number of households
First, we need to find the total number of households. We sum up the number of households for each number of dogs.
Total households $N=1491 + 425+168 + 48+29+14$
$N = 1491+425 = 1916$; $1916 + 168=2084$; $2084+48 = 2132$; $2132+29 = 2161$; $2161+14=2175$

### Step 2: Calculate the probability for each number of dogs
The probability $P(x)$ for a discrete random variable is given by $P(x)=\frac{	ext{Frequency of }x}{	ext{Total frequency}}$.
- For $x = 0$: $P(0)=\frac{1491}{2175}\approx0.6855$
- For $x = 1$: $P(1)=\frac{425}{2175}\approx0.1954$
- For $x = 2$: $P(2)=\frac{168}{2175}\approx0.0772$
- For $x = 3$: $P(3)=\frac{48}{2175}\approx0.0221$
- For $x = 4$: $P(4)=\frac{29}{2175}\approx0.0133$
- For $x = 5$: $P(5)=\frac{14}{2175}\approx0.0064$

### Step 3: Calculate the mean ($\mu$) of the probability distribution
The formula for the mean of a discrete probability distribution is $\mu=\sum_{i}x_{i}P(x_{i})$
$\mu=(0	imes0.6855)+(1	imes0.1954)+(2	imes0.0772)+(3	imes0.0221)+(4	imes0.0133)+(5	imes0.0064)$
$=0 + 0.1954+0.1544 + 0.0663+0.0532+0.032$
$=0.1954+0.1544=0.3498$; $0.3498 + 0.0663=0.4161$; $0.4161+0.0532 = 0.4693$; $0.4693+0.032=0.5013$

### Step 4: Calculate the variance ($\sigma^{2}$) and then the standard deviation ($\sigma$)
The formula for variance is $\sigma^{2}=\sum_{i}(x_{i}-\mu)^{2}P(x_{i})$
- For $x = 0$: $(0 - 0.5013)^{2}	imes0.6855=( - 0.5013)^{2}	imes0.6855\approx0.2513	imes0.6855\approx0.1723$
- For $x = 1$: $(1 - 0.5013)^{2}	imes0.1954=(0.4987)^{2}	imes0.1954\approx0.2487	imes0.1954\approx0.0486$
- For $x = 2$: $(2 - 0.5013)^{2}	imes0.0772=(1.4987)^{2}	imes0.0772\approx2.2461	imes0.0772\approx0.1734$
- For $x = 3$: $(3 - 0.5013)^{2}	imes0.0221=(2.4987)^{2}	imes0.0221\approx6.2435	imes0.0221\approx0.1379$
- For $x = 4$: $(4 - 0.5013)^{2}	imes0.0133=(3.4987)^{2}	imes0.0133\approx12.2410	imes0.0133\approx0.1628$
- For $x = 5$: $(5 - 0.5013)^{2}	imes0.0064=(4.4987)^{2}	imes0.0064\approx20.2383	imes0.0064\approx0.1295$

Sum of these values for variance:
$0.1723+0.0486 = 0.2209$; $0.2209+0.1734 = 0.3943$; $0.3943+0.1379 = 0.5322$; $0.5322+0.1628 = 0.695$; $0.695+0.1295 = 0.8245$

Standard deviation $\sigma=\sqrt{\sigma^{2}}=\sqrt{0.8245}\approx0.908$

# Probability Distribution Table:
| Dogs ($x$) | 0 | 1 | 2 | 3 | 4 | 5 |
| --- | --- | --- | --- | --- | --- | --- |
| Probability ($P(x)$) | $\approx0.6855$ | $\approx0.1954$ | $\approx0.0772$ | $\approx0.0221$ | $\approx0.0133$ | $\approx0.0064$ |

# Mean:
$\approx0.50$ (rounded to two decimal places)

# Standard Deviation:
$\approx0.91$ (rounded to two decimal places)

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