Chpt 5 - Discrete Random Variables

Question 1

What is a random variable?

Answer 1

A variable whose possible values are:
-Numerical
-Depend on chance

Question 2

What are the 2 types of random variables?

Answer 2

Discrete random variable

Continuous random variable

Question 3

What is a variable whose possible values are numerical and are depending on chance?

Answer 3

Random variable

Question 4

What is a discrete random variable?

Answer 4

A random variable whose possible values can be listed

(I understand this as whole numbers)

Question 5

How do we show the distribution of a random variable?

Answer 5

We can use a graph, table, or formula to show:
-possible values
-relative frequency

(so what and how often)

Question 6

What do we use to denote a random variable and the values that it can take

Answer 6

X - random variable

x - possible values the variable can take

Question 7

What does P(X = 1) mean?

Answer 7

The probability of the variable being equal to 1

So 1 email address, 1 child at home, has 1 pet etc.

Question 8

What does P(X < 2) mean?

Answer 8

The probability of the variable being less than 2

So 1 or no kids at home, maybe 1 or no pets for example

Question 9

What is the value of:

P(X = x)

Answer 9

0 - 1

Because it is a probability

Question 10

How do we denote the probability of a value of a random variable?

Answer 10

P(X = x)

P is probability

X is the random variable

x is the value the variable can take that we are looking at

Question 11

What is the sum of all the probabilities of a random variable?

Answer 11

1

Each individual value will be between 0-1 as it is probabilities of the entire population. The entire population counted together is 1 (meaning its 100%)

Question 12

Distribution of a value is:

x -1 0 1
X 0.5 0.3 0.4

Is this distribution correct? Why/why not?

Answer 12

No, because the sum is greater than 1

Question 13

You have a distribution table as follows:

x 0 1 2
X 0.4 0.2 a

What should the value of a be?

Answer 13

a = 1 - 0.4 - 0.2 = 0.4

Question 14

How do we describe the center when discuss the distribution of a discrete random variable?

Answer 14

Population means - but only if we have the population data available, if not, we use relative frequencies

Question 15

What is the formula for determining the population mean of a discrete random variable X?

Answer 15

μ = ∑ xP(X=x)

So we multiply each value (x) by it’s own probability (P(X=x)) and then add up all of these to create the average, or mean (mu)

Question 16

What are the steps to determining the population mean of a discrete random variable?

Answer 16

μ = ∑ xP(X=x)

1. Multiply each possible value (x) and it’s corresponding probability ( P(X=x) )
2. Add all the values in 1 up to determine the mean

Question 17

We learned previously that

μ = (x1+x2+x3+…+xn) / N

Is how we determine the mean.

Why do we not divide by N in the formula used to find the mean of a discrete random variable as seen here:

μ = ∑ xP(X=x)

Answer 17

It’s about probability. To use the discrete random variable in

μ = ∑ xP(X=x)

equation, we multiply each value by it’s probability. Probability is found using:

P =f/N

So we have already divided by N before getting to the equation.

Question 18

We did a survey looking at people who had dogs (X). If the probability of people having one dog (X=1) is 0.25, what is the value of xP(X=1) ?

Answer 18

xP(X=1) = 1 x 0.25 = 0.25

Question 19

We did a survey of homes in Edmonton, and one of the variables was how many dogs did they have. How would we denote this variable?

Answer 19

A capital letter, such as A or X etc.

Any value we assign to that, such as 3 dogs, would use the lower case of the same letter so a or x etc.

Question 20

We did a survey of homes in Edmonton, and one of the variables was how many dogs did they have. Now we are determining how many people who had 3 dogs. How would we denote the value of 3 dogs?

Answer 20

A lower case letter such as a or x etc.

You just have to make sure that it is the same letter, just a different case, as what you use to denote the variable.

So A is the number of dogs that a random family has (variable), and a is 3 dogs (value)

Question 21

We did a survey of homes in Edmonton, and one of the variables was how many dogs did they have. What is the probability that a family has 3 dogs?

x 1 2 3 4
P(X=x) 0.2 0.4 0.3 0.1

Answer 21

P(X=3) = 0.3

Question 22

We did a survey of homes in Edmonton, and one of the variables was how many dogs did they have. What is the xP(X=x) value of a family that has 3 dogs?

x 1 2 3 4
P(X=x) 0.2 0.4 0.3 0.1

Answer 22

xP(X=3)

3 x 0.3 = 0.9

Question 23

We did a survey of homes in Edmonton, and one of the variables was how many dogs did they have. What is the mean for this discrete random variable?

x 0 1 2 3
P(X=x) 0.1 0.2 0.4 0.3

Answer 23

Step 1: Find x(PX=x) values

0 x 0.1 =0
1 x 0.2 = 0.2
2 x 0.4 = 0.8
3 x 0.3 = 0.9

Step 2: Find Sum of all

0 + 0.2 + 0.8 + 0.9 = 1.9

Mean is 1.9 dogs per home

Question 24

You are playing Monopoly with Dani for $1 each game. The probability that you lose to Dani is 0.3, the probability that you win over Dani is 0.2, the probability that Dani gets mad and flips the table rendering no one a winner is 0.5.

How much money would I win or lose on average in a game?

What if we played 100 games?

Answer 24

X: times I win
x = 1 : I win
x = 0 : Dani flipped the table
x = -1 : Dani beat your ass

x P(X=x) xP(X=x)
1 0.2 0.2
0 0.5 0
-1 0.3 -0.3

μ=∑xP(x)
= 0.2 + 0 - 0.3
= -0.1

The average outcome is that I lose $0.10 per game

100 x -0.1 = -10
So in 100 games, I would lose $10…and my sanity

Question 25

What is the formula for population standard deviation of a discrete random variable?

Answer 25

σ = √ ∑ (x-μ)squared X P(X=x)

So multiply the (x-μ) squared column by the P(X=x) value. Add those all up. Square root the total

Question 26

What are the steps for determining the population standard deviation of a discrete random variable?

Answer 26

σ = √ ∑ (x-μ)squared X P(X=x)

Step 1: find μ, the mean of the discrete random variable

Step 2: Find (x-μ) squared time P(X=x) for all of the possible values for this discrete random variable

Step 3: Add all multiplications in step 2

Step 4: Take the square root of the sum in step 3

Question 27

What is the variance of a random variable?

What is used to determine the varience?

Answer 27

The square of standard deviation, also used to describe the spread.

Its the ∑ (x-μ)squared X P(X=x) value. This goes under a square root to get the standard deviation

Question 28

What are the properties of Bernoulli trials?

Answer 28

• each trial has 2 outcomes
  success (s) and failure (f)
• trials are independent
• probability of success remains the same from trial to trail

Question 29

What is the probability of success denoted by in Bernoulli trials?

Answer 29

lower case p

Question 30

Is a success outcome in a Bernoulli trial a good thing?

Answer 30

A success outcome is not necessarily “good” in the everyday sense of the word. Success is merely a name for 1 of the 2 possible outcomes on a single trial

Question 31

How are the results of a Bernoulli trial denoted?

Answer 31

s - success
f - failure

Question 32

What is the number of successes in a sequence of MANY Bernoulli trials called?

Answer 32

Binomial random variable

Question 33

What is a binomial random variable?

Answer 33

The number of successes in a sequence of MANY (n) Bernoulli trials

Question 34

Is the following a Bernoulli trial? Why/why not?

“The number of heads when flipping 2 balanced coins”

Answer 34

Yes because it is the number of successes (heads) in 2 Bernoulli trials.

It is a random variable because it is numerical and depends on chance

Question 35

Is the following a Bernoulli trial? Why/why not? If possible, identify n, s, f and p.

The number of boys that a couple will have if they plan to have 5 children

Answer 35

yes
- 2 possible outcomes
- independent
- p remains the same between trials

n=5

s={boy}; f={girl}

p = 0.5

Question 36

Is the following a Bernoulli trial? Why/why not? If possible, identify n, s, f and p.

The number of times that 4 is observed when rolling a balanced die 6 times.

Answer 36

yes
- 2 possible outcomes
- independent
- p remains the same between trials

n=6

s={4}; f={1, 2, 3, 5, 6l}

p = 1/6

Question 37

Is the following a Bernoulli trial? Why/why not? If possible, identify n, s, f and p.

There are 10 toys: 6 cars and 4 planes. A boy randomly picks 3 toys without replacement. Is “the number of cars among the 3 picked toys” a binomial random variable?

Answer 37

No because it doesn’t meet all the criteria

2 possible outcomes - yes there is car and plane

independent - no, choosing the first toy will affect the options for the second toy

probability stays the same between trials - no because the cars are not put back once picked out, decreasing the number of cars/toys available

For example:
First pick there is 6/10 chance for a car and a 4/10 chance of a plane. Say a car is selected. On the second pick there is now 5/9 chance for a car and 4/9 chance for a plane.

Question 38

What are the to important parameters that determine the distribution of a binomial random variable?

Answer 38

-number of Bernoulli trials (n)

-success probability (p)

Question 39

What are the steps for determining the distribution of a binomial random variable?

Answer 39

Step 1: Determine the number of Bernoulli trials (n)

Step 2: Find the value (x) desired

Step 3: Determine the success probability (p) for each trial

Step 4: Find nChoosex

Step 5: Find p to the power of x

Step 6: Find 1-p to the power of n-x

Step 7: Multiply answers to steps 4, 5, and 6

Question 40

How do we determine the probability of a binomial random variable when we are looking for multiple values?

Answer 40

Determine the probability of each value (using the P(Y=y) equation, and then add them together.

If there are many values, such as we are looking for >1 out of 10 trials, determine the probability of each value (using the P(Y=y) that we don’t want, such as 0 and 1 in this example, and subtract those from 1
(Remember a probability of 1 is all of the values in a population, so subtract what you don’t needs leads you with what you want)

Question 41

How do you determine the mean of a binomial variable?

Answer 41

μ = np

Question 42

How do you determine the standard deviation of a binomial variable?

Answer 42

σ = √np(1-p)