Random Variables
What is a continuous random variable?
A random variable that can assume any value contained in one or more intervals
What is the probability distribution?
a list with all the possible values that the random variable can assume and their corresponding probabilities
What is the equation of the mean of a discrete random variable x
μ = ΣxP(x)
Name two of the conditions a binomial experiment must have
1. There are n identical trials
2. Each trial has only two possible outcomes.
3. The probability of the two outcomes remain constant
4. The trials are independent
What is the formula for the hypergeometric probability distribution
Where
N= total number of elements in the population
r= number of successes in the population
N-r= number of failures in the population
n= number of trials
x= number of successes in n trials
n-x= number of failures in n trials
rCx N-rCn-x
NCn
Discrete or continuous?
The weight of a letter
Is this a valid probability distribution?
X P(x)
7 .7
8 .5
9 -.2
Although the sum of al probabilities listed in this table is equal to 1.0, one of the probabilities is negative. This violates the first condition of a probability distribution. Therefore, this table does not represent a valid probability distribution.
x P(x)
0 .15
1 .20
2 .35
3 .30
1.80
Which of the following are binomial experiments?
a. Rolling a die many times and observing the number of spots
b. Rolling a die many times and observing whether the number obtained is even or odd
c. Selecting a few voters from a very large population of voters and observing whether or not each one of them favors a certain proposition in an election when 54% of all voters are known to be in favor
Only b
Name two conditions to apply the poisson probability distribution
1. x is a discrete random variable
2. The occurrences are random
3. The occurrences are independent
Discrete or Continuous?
The price of a house
Continuous
Given this table what is the probability that x assumes a value less than 3?
X P(x)
0 .03
1 .17
2 .22
3 .31
4 .15
5 .12
.42 or 42%
Let x be the number of heads obtained in two tosses of a coin. The following tables lists the probability distribution of x. Calculate the mean and standard deviation of x
X P(x)
0 .25
1 .50
2 .25
μ=1.00
σ=.707
What is the formula for the mean and standard distribution of a binomial distribution?
μ=np
σ=√(npq)
An internal revenue service inspector is to select 3 corporations from a list of 15 for tax audit purposes. Of the 15 corporations, 6 earned profits and 9 incurred losses during the year for which the tax returns are to be audited. If the IRS inspector decides to select 2 corporations randomly, find the probability that the number of corporations in these 3 that incurred losses during the year for which the tax returns are to be audited is
a. exactly 2
b. none
c. at most 1
a. .4747
b. .0440
c. .3407
A household can watch National news on any of the three networks- ABC, CBS, or NBC. On a certain day, five households randomly and independently decide which channel to watch. Let x be the number of households among these five that decide to watch news on ABC. Is x a discrete or a continuous random variable? Explain. What are the possible values that x can assume?
x is discrete. It can be 0,1,2 or 3
Which of the following is NOT a characteristic of a probability Distribution?
1. 0≤P(x)≤1
2. 0≥P(x)≥1
3. ΣP(x)=1
2
What is the formula for the standard deviation of a discrete random variable?
σ=√(Σ[(x-μ)2(P(x)])
Let x be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities.
a. P(x=5) for n=8 and p=.7
b. P(x=3) for n=4 and p=.4
c. P(x=2) for n=10 and p=.6
a. .2541
b. .1536
c. .3241
On average, 5.4 shoplifting incidents occur per week at an eectronics store. Find the probability that exactly 3 such incidents will occur during a given week at this store.
.1185
5 cars stop at a gas station. They can either fill their tank 25%, 78%, or 99.9%. Let x be the number of cars that fill their tank 99.9% of the way. Is x a discrete or continuous variable?
Discrete
According to the most recent data from the Insurance Research Council, 16.1% of motorists in the United States were uninsured in 2010. Suppose that current 16.1% of motorists in the United States are uninsured. Suppose that two motorists are selected at random. Let x denote the number of motorists in this sample of two who are uninsured. Construct the probability distribution table of x.
X P(x)
0 .7039
1 .2702
2 .0259
A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are $10 million from the office building, $5 million from the theater, and $2 million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are .15, .30, .45, and .10, respectively. Let x be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of x and find the mean and standard deviation.
σ= $3.015 million
μ= $3.9 million
A fast food chain store conducted a taste survey before marketing a new hamburger. The results of the survey showed that 70% of the people who tried this hamburger liked it. Encouraged by this result, the company decided to market the new hamburger. Assume that 70% of all people like this hamburger. On a certain day, eight customers bought it for the first time.
a. Let x denote the number of customers in this sample of eight who will like this hamburger. Using the binomial probabilities rules, obtain the probability distribution of x and draw a graph of the distribution. Determine the mean and standard deviation of x.
b. Using the probability distribution of part a, find the probability that exactly three of the eight customers will like this hamburger.
a. μ = 5.6000
σ = 1.296
b. .0467
Let x be a Poisson random variable. Using the Poisson probabilities table, write the probability distribution of x for each of the following. Find the mean and standard deviation for each of these probability distributions.
a. λ= 1.3
b. λ=2.1
a. μ= 1.3
σ= 1.140
b. μ= 2.1
σ= 1.449