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title: Stat05
tags: [Stat]

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title: Stat05
tags: Stat
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# Chapter 5: Discrete Probability Distributions


## 5.1 Discrete random variables and their probability distributions. 


### Exercise

#### 5.1.17-5.1.18


#### 5.1.23 A pair of Dice

Toss a pair of dice and record $x$, the sum of the numbers on the two upper faces. 

Find the probability distribution, and the probability distribution, and calculate its mean and variance. 

#### 5.1.32 Fire Alarms (Related to the Binomial random variable)

A fire-detection device uses three temperature-sensitive cells action independently of one another so that any one or more can activate the alarm. Each cell has a probability of 0.8 of activating the alarm when the temperature reaches 57$^\circ$C or higher. Let $X$ equal the number of cells activationg the alarm when the temperature reaches 57$^\circ$C. 

a. Find the probability distribution of $X$.
b. Find the probability that the alarm will function when the temperature reaches 57$^\circ$C.
c. Find the expected value and the variance for the random variable $x$. 

#### Sol.

After 5.2, it is easy to identify tat $X \sim Binomial(n=3, p=0.8)$



## 5.2 Binomial Probability Distribution

### Key ideas

Notation: $p+q=1$, $q=1-p$. 

1. $X\sim Binomial(n, p)$, if
$$P(X=x) = C^{n}_x p^{x}(1-p)^{(n-x)},$$for $x=0, 1,\ldots,n$. It has the mean: $\mu=np$, variance: $\sigma^2=npq$, and standard deviation: $\sigma=\sqrt{npq}$. 
2. Use a calculator and the probability distribution formula to find the answer.:cry:  
3. Use Table 1 (p.682-p.687) to find the probabilities.:smile:

### Exercise


#### 5.2.2-6

#### 5.2.30
a. $P(X<12)=P(X\leq 11) = 0.748$ for $n= 20$, p=0.5. 

e. $P(3<x<7)=P(X\leq 6)-P(X\leq 2)$ for $n=10$ $p=0.5$

Be careful you know how to get the left-, right- tailed probability, and the probabiltiy of an interval.  


#### 5.2.57 Men's BFF

According to the Human Society of America, there are approximately 77.8 million owned dogs in the United States, and approximately 50% of dog-owning households have small dogs. Suppose the 50% figure is correct and that $n=15$ dog-owning households are randomly selected for a pet ownership survey. 

a. What is the probability that exactly eight of the households have small dogs?
b. What is the probability that at most four of the households have small dogs?
c. What is the probability that more than 10 households have small dogs? 


## 5.3 Poisson Distribution

### Key ideas

Notation: $X\sim Poisson(\mu)$ 

1. $X\sim Poisson(\mu)$, if
$$P(X=k) = e^{-\mu} \frac{\mu^k}{k!},$$for $k=0, 1,\ldots,\infty$. It has the mean: $\mu$, variance: $\sigma^2=\mu$, and standard deviation: $\sigma=\sqrt{\mu}$. 
2. Use a calculator and the probability distribution formula to find the answer.:cry:
3. Use Table 2 (p.688-p.689) to find the probabilities.:smile:


### Exercise

#### Basics: 5.3.3

$\mu=2.5$, 

$P(X=1)$

$P(X\leq 2)$



#### 5.3.20 Accident Prone

According to a study conducted by the Department of Pediatrics at the University of California, San Fransico, children who are injured two or mote times tend to sustain these injuries during a relatively limited time, usually 1 year or less. If the average number of injuries per year for school-aged children is two, what are the probabilities of these events? 

a. A school-age child will sustain two injuries during the year. 
b. A school-age child will sustain two or more injuries audring the year. 
c. A school-age child will sustain at most one injury druing the year. 













