Chpt 5 - Discrete Random Variables Flashcards by Amanda Butt

What is a random variable?

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

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What are the 2 types of random variables?

Discrete random variable

Continuous random variable

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

Random variable

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What is a discrete random variable?

A random variable whose possible values can be listed

(I understand this as whole numbers)

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How do we show the distribution of a random variable?

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

(so what and how often)

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

X - random variable

x - possible values the variable can take

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What does P(X = 1) mean?

The probability of the variable being equal to 1

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

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What does P(X < 2) mean?

The probability of the variable being less than 2

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

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What is the value of:

P(X = x)

0 - 1

Because it is a probability

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

P(X = x)

P is probability

X is the random variable

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

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

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%)

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Distribution of a value is:

x -1 0 1
X 0.5 0.3 0.4

Is this distribution correct? Why/why not?

No, because the sum is greater than 1

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You have a distribution table as follows:

x 0 1 2
X 0.4 0.2 a

What should the value of a be?

a = 1 - 0.4 - 0.2 = 0.4

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

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

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

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)

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

μ = ∑ xP(X=x)

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

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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)

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.

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) ?

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

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?

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.

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?

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)

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

P(X=3) = 0.3

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

xP(X=3)

3 x 0.3 = 0.9

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

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

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?

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

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

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

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

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

What is the variance of a random variable? What is used to determine the varience?

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

What are the properties of Bernoulli trials?

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

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

lower case p

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

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

How are the results of a Bernoulli trial denoted?

s - success f - failure

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

Binomial random variable

What is a binomial random variable?

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

Is the following a Bernoulli trial? Why/why not? "The number of heads when flipping 2 balanced coins"

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

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

yes - 2 possible outcomes - independent - p remains the same between trials n=5 s={boy}; f={girl} p = 0.5

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.

yes - 2 possible outcomes - independent - p remains the same between trials n=6 s={4}; f={1, 2, 3, 5, 6l} p = 1/6

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?

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.

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

-number of Bernoulli trials (n) -success probability (p)

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

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

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

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)

How do you determine the mean of a binomial variable?

μ = np

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

σ = √np(1-p)