Random Variables

Question 1

Random Variable

Answer 1

A random variable is the numerical outcome of a random experiment.

Question 2

Example of a random experiment

Answer 2

1. Imagine a machine that draws a student at random from the student body. (The experiment). The student’s height, weight, sex are all random variables.
2. The outcome of the toss of a coin is a random variable.
3. Toss a pair of dice. Let X represent the sum of the dots on the two dice.

Question 3

Notation for a random variable

Answer 3

1. The random variable is typically named X
2. The probability that a random variable X has the value x is written as Pr(X=x) or simply p(x)

Question 4

Let X be the random variable representing the outcome of throwing a pair of dice and summing them. If we draw a probability histogram, what value will it be centered around and what will its probability be?

Answer 4

7

p(7) = 1/6

Question 5

What principle enables us to use the properties of theoretical distributions to model data?

Answer 5

We can think of the frequentist definition of probability: ie: the relative frequence of an event “in the long run.” If a particular random experiment is repeated many times, the outcome begins to look like the theoretical distribution.

Question 6

What symbols are coventionally used for the statistical properties of population versus sample data?

Answer 6

Population mean: µ

Sample mean: x̅

Population standard deviation: σ

Sample standard deviation: s

Population proportion: p

Sample proportion: p̂

Question 7

Mean of a random discrete variable

Answer 7

µ = ∑ x·p(x)

The sum of all the possible values, each weighted by its probability.

Question 8

A pair of coins is tossed multiple times. At each trial the number of heads is counted (ie 0, 1 or 2 heads). Let x represent the value. The following results are obtained:

x n_x

0 260

1 517

2 223

Calculate the mean and compare it to the expected mean.

Answer 8

x̅ = 0*(.26) + 1*(.517) + 2*(.223)

= .963

Model µ = 0*.25 + .5*.5 + .25*.5

= 1

Question 9

Variance of a random discrete variable

Answer 9

σ^{<span>2</span>} = ∑ (x-µ)² p(x)

Question 10

What is the relationship between the mean and expected value of a random variable?

Answer 10

The mean of a random variable is equal to the expected value of the variable.

µ = E[X]

Question 11

What is the probability that a random continuous variable will be equal to a particular value?

Answer 11

0

Since the random variable is continous, it can take on an infinite number of values. Since all these values are equally likely, the probability that it is a particular value will be 0.

For example, consider a balanced, spinning pointer. The probability that it will stop within a particular range of values can be calculated (for example P(.25 ≤ X ≤ .75) = .5) but not for a particular value such as .5.

Question 12

X is a continuous random variable with probability density function given by f(x) = cx for 0 ≤ x ≤ 1, where c is a constant. Find c.

Answer 12

c=2

For any continuous random variable with probability density function f(x), we have that: ∫ f(x) dx= 1 when we integrate over all possible values of x.

We know that cdf If we integrate f(x) between 0 and 1 we get c/2. Hence c/2 = 1 (from the useful fact above!), giving c = 2.

Question 13

Let x be a random variable and f(x) be its probability density function. What is the probability that x will occur in the interval [a, b]

Answer 13

∫f(x) dx, definite integral from a to b

Question 14

If f(x) is the probability density function of a random continous variable X, what is the mean and variance ?

Answer 14

• µ = ∫ x·f(x) dx* from negative infinity to positive infinity (or all possible values of x).
• σ² = ∫ (x-µ)²·f(x) dx* from negative infinity to positive infinity (or all possible values of x).

Question 15

Suppose we are playing a gambling game depending on a coin toss. You ante up $6 to play; you get $10 if the result is a head. $0 if it is a tail.

What is your expected winning?

Answer 15

-1

Let W be a random variable representing you winnings:

W = 10X -6

where X is the original random variable with value 1 when a head results.

E[W] = E[10X - 6] = 10*E{X} -6 = -1

Question 16

Suppose we are playing a gambling game depending on a coin toss. You ante up $6 to play; you get $10 if the result is a head. $0 if it is a tail.

What is the variance of your expected winning?

Answer 16

25

σ²(W) = σ²(10X +6)

= 100σ²(X)

= 25

so:

σ(X) = 5

Question 17

Suppose we toss two coins. The value of a head is 1. What is the variance of the outcome?

Answer 17

.5

Let X₁ and X₂ be the two random variables. When two random variables are independent, their variances will add. So:

σ^₂(X₁+X₂) = σ²(X₁) + σ2(X₂)

= .25 + .25

= .5

Question 18

Expectation of the sum of many random variables:

E[∑x_i] =?

Answer 18

E[∑x_i] =∑E[x_i]

The expectation of the sum of many random variables is equal to the sum of individual expectations.

(What does this mean?)