Probability Distrubution Flashcards
(25 cards)
What is a Random Variable?
A Random Variable X associates a unique numerical value with each outcome in the sample space
The value of X is denoted by lower case x
What is the range of a Random Variable X?
The set of values that X can take
This set is referred to as the range of X
What two features do random variables require?
- Numerical function
- Range
To be full defined
What type of variable is X if it can take values 0 or 1?
Discrete random variable
Example: X=1 if a test result is positive, X=0 if negative
What is the range of the random variable Y, defined as ‘sum of scores when two dice are thrown’?
The set of any integer in (2,3,4,…,12)
What type of random variable is Y when two dice are thrown?
Y is a discrete random variable since its range is finite
The random variable Z is defined as ‘time between two red cars on the Central Motorway on a Sunday’.
The range of Z is [0,7hrs].
What type of random variable is Z?
The range of Z is a close interval [0,7hrs]. This indicates that Z is a continuous random variable
Are random variables discrete or continuous?
Either
What is a discrete random variable?
A variable that can take on a countable number of distinct values.
Define f(x) in the context of discrete random variables.
f(x) = P(X=x), the probability function of the discrete random variable X.
What does P(X=x) represent?
The probability that the random variable X is equal to x.
Where X is the set and x is the value
What is the range of probabilities for discrete random variables?
Probabilities can only be between 0% and 100%.
Or 0 and 1
What is the sum of the probability functions for all values of X?
The sum is equal to 1.
P(X=a) + P(X=b) + P(X=c) = 1
f(0.2) + f(0.5) + f(0.3) = 1
What is the cumulative distribution function (CDF) F(x)?
F(x) is defined as F(x) = P(X ≤ x).
f(3) = P(Y<= 3) = P(Y=2) + P(Y=3)
How is the expected value of a discrete random variable X denoted and how else can it be referred to?
E(X) or μ.
The mean of X
What is the formula for calculating the expected value E(X)?
Is this the same for all random variables?
E(X) = μ = Σ [x * P(X = x)].
No replace the sim with the integral from infinity to negative for continuous random variables
What does P(X=-1)=0.97 indicate in a probability scenario?
There is a 97% chance of the -1 outcome occurring.
Eg. Lose £1
P(x=-1)=0.97
P(x=24)=0.02
P(x=48)=0.01
In the given game scenario, what are the probabilities of winning £24 and £48?
2% chance of 24 and 1% chance of 48.
P(x=-1)=0.97
P(x=24)=0.02
P(x=48)=0.01
What is the mean for the game described?
E(X) = μ = Σ [x * P(X = x)] = (-1)(0.97) + (24)(0.02) + (48)*(0.01) = -0.01.
For a fair game. The average outcome would be 0
True or False: For a fair game, the average return should be 0.
True.
Fill in the blank: The cumulative distribution function F(X) is defined as ______.
F(X) = P(X ≤ x)
What is an example of a probability function outcome?
P(X=a) + P(X=b) + … + P(X=n) = 1.
What does the notation E(Y) represent?
The expected value of the random variable Y
What is the significance of the expected value E(X) in decision-making and why might it not be the best indicator?
It helps in determining whether to engage in a game based on potential returns. However it is not as good of an indicator as variability due to h9w easily it is effected by outliers
