Topic 3 Flashcards
What is a random variable and how is it denoted?
- a rule that turns the outcome of a random experiment into a real number
- X
What are the two types of random variables?
- discrete
- continuous
What is a discrete random variable?
- a variable that only takes on a countable number of distinct values, such as 0, 1, 2, 3, 4
What is the probability distribution of a discrete random variable?
- a list of probabilities associated with each of its possible values
To compute?
- the mean of a discrete random variable
To compute?
- the variance of a discrete random variable
To compute?
- the standard deviation of a discrete random variable
Consider the following probability distribution for variable X, the number of spam emails received in a given day.
(a) Compute the mean value, E(X).
(b) Compute the standard deviation of X.
(c) Interpret the results in (a) and (b).
(d) What is the probability that exactly 4 emails are received in a given day?
(e) What is the probability that at least 3 emails are received in a given day?
(f) What is the probability that no more than 3 emails are received in a given day?
(g) What is the probability that more than 5 emails are received in a given day? Explain your answer.
c) On a given day, the expected number of spam e-mails received is 3, with a standard deviation of 0.89.
d) P(X=4) = 0.20
e) P(3≤X≤5) = 0.5 + 0.2 + 0.05 = 0.75
f) P(1≤X≤3) = 0.05 + 0.2 + 0.50 = 0.75
g) 0 - The probability distribution for the random variable X provides five mutually exclusive and collectively exhaustive outcomes. In fact, the sum of the probabilities for these outcomes is already 1.
What is a binomial random variable and how is it denoted?
- an example of a discrete random variable where there are only two outcomes from “n” independent trials
- the probability of a “success” is the same for each trial, and this is denoted as π
- the probability of “failure” is 1 - π
To compute?
- the mean of a binomial random variable
To compute?
- the variance of a binomial random variable
To compute?
- the standard deviation of a binomial random variable
Suppose that 8 per cent of the loans made by the YourBank are never repaid. In a random sample of 20 loans:
(a) Do you think this is an example of a binomial random variable? Justify your answer.
This is an example of a binomial random variable.
The experiment has only two possible outcomes, “loans are never repaid” or “loans are repaid”.
We have fixed number of independent trials, which are 20 observations.
The probability that loans are never repaid is 0.08, and this is the same for all 20 loans.
If X is the number of loans that are never repaid, then,
X∼b (20, 0.08)
Suppose that 8 per cent of the loans made by the YourBank are never repaid. In a random sample of 20 loans:
(b) Using the PHStat output, what is the probability that:
1. Exactly two are not repaid?
2. Less than four are not repaid?
3. More than four are not repaid?
4. Either two or three are not repaid?
5. Exactly 15 are repaid?
- P(X=2) = 0.271091
- P(X<4) = P(X<=3) = 0.929385
- P(X>4) = 1- P(X<=4) = 1- 0.981656 = 0.018344
- P(X=2 OR X=3) = 0.271091 + 0.141439 = 0.41253
- P(X’=15) = P(X=5) = 0.014545
Suppose that 8 per cent of the loans made by the YourBank are never repaid. In a random sample of 20 loans:
(c) The mean of the number of loans that are not repaid is:
μ = nπ = 20 x 0.08 = 1.6
Suppose that 8 per cent of the loans made by the YourBank are never repaid. In a random sample of 20 loans:
(d) The standard deviation of the number of loans that are not repaid is:
What is a continuous random variable?
- one which takes an infinite number of possible values
- usually measurements
- not defined at specific values
- defined over an interval of values, and is represented by the area under a curve
- the probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite
What is a normal distribution?
- a bell-shaped density curve described by its mean, μ, and standard deviation, σ
What shape is the density curve of a normal distribution?
- symmetrical, centred about its mean, with its spread determined by its standard deviation
Why do we need to standardize the values of a normal distribution?
- the area under the curve is not easy to calculate for a normal random variable X with mean μ and standard deviation σ
- hence, we need to standardize the values, which is computed from X by subtracting μ and dividing by σ
What is the mean and standard deviation of the standardized normal values and what are they denoted by?
- mean 0 and standard deviation of 1
- Z
How do we obtain probabilities of X from a normal distribution with mean, μ, and standard deviation, σ?
- we first need to convert value of X using the transformation formula:
- draw a diagram and shade the appropriate area under the curve
- read the appropriate value from Table E2
To compute?
- the transformation formula to convert to value of X to the standardized value
(to obtain probabilities of X from a normal distribution with mean, μ, and standard deviation, σ)
Given a standardized normal distribution with a mean of 0 and a standard deviation of 1 (as in Table E.2),
(a) What is the probability that Z is less than -1.25?
(b) What is the probability that Z is less than 1.25?
(c) What is the probability that Z is between -1.25 and 1.25?
(d) What is the value of Z where only 2.5% of all possible Z values are larger? (P(Z > Z) 0.025)
a) P(Z < -1.25) = 0.1056
b) P(Z < 1.25) = 0.8944
c) P(-1.25 < Z < 1.25) = P(0.8944-0.1056) = 0.7888
d) The probability for 100% - 2.5% = 97.5% = 0.9750 = Z score of +1.96
Given a normal random variable with mean, μ = 50 and σ = 4
(a) What is the probability that X > 40?
(b) What is the probability that X < 45?
(c) What is the probability that X is between 40 and 60?
(d) 5% of the values are less than what unknown value of X? (P(X < Xa) = 0.05)
The number of shares traded daily on the New York Stock Exchange (NYSE) is referred to as the volume of trading. Assume that the number of shares traded on the NYSE is a normally distributed random variable, with a mean of 1.8 billion and a standard deviation of 0.15 billion.
For a randomly selected day, what is the probability that the volume is:
(a) above 2 billion?
(b) below 1.7 billion?
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
Use Table E.2 from your text to answer the following questions.
You will need to use a calculator.
Enter z-scores to one or two decimal places as appropriate.
Enter probabilities to 4 decimal places and include the leading 0.
For example: 0.1234
