What is a discrete random variable ?

A discrete random variable is a variable that can only take certain values within a set. Often this involves counting something (for example, the number of heads when a coin is tossed 10 times).

What is a probability distribution ?

A probability distribution is a description of all possible values a random variable can take, along with their associated probabilities .

True or False? Random variables are denoted using lower case letters.

False. Random variables are denoted using upper case letters.

What is a discrete uniform distribution ?

A discrete uniform distribution is a probability distribution where a random variable takes a finite number of values, and each value has an equal probability of occurring.

What is the sum of probabilities for all values of a discrete random variable?

The sum of probabilities for all values of a discrete random variable is 1.

True or False? A discrete random variable can only take a finite number of values.

False. A discrete random variable can take a finite or infinite number of values.

Define expected value .

Expected value is the average outcome of an experiment if it is repeated many times . It is calculated as the sum of each possible outcome multiplied by its probability.

True or False? The expected value of a random variable is always a whole number .

False. The expected value of a random variable can be any real number , including fractions or decimals .

How is the expected gain or loss typically calculated in a game?

The expected gain or loss for a game is typically calculated by subtracting the cost to play the game from the expected value of the prize .

What is special about the mean of a symmetrical distribution ?

The mean of a symmetrical distribution is the same as its median .

IBMathsDPAnalysis & Approaches (AA)HLFlashcards4. Statistics & ProbabilityDiscrete Random Variables

Discrete Random Variables (DP IB Analysis & Approaches (AA))

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What is a discrete random variable?
A discrete random variable is a variable that can only take certain values within a set.
Often this involves counting something (for example, the number of heads when a coin is tossed 10 times).
What is a probability distribution?
A probability distribution is a description of all possible values a random variable can take, along with their associated probabilities.
True or False?
Random variables are denoted using lower case letters.
False.
Random variables are denoted using upper case letters.
What does the notation $P (X = x)$ mean?
$P (X = x)$ is the probability that the random variable $X$ takes the value $x$ .
E.g. $P (X = 3)$ is the probability that the random variable $X$ takes the value 3.
What is a discrete uniform distribution?
A discrete uniform distribution is a probability distribution where a random variable takes a finite number of values, and each value has an equal probability of occurring.
What is the probability of each value occurring in a discrete uniform distribution with $n$ values?
The probability of each value occurring in a discrete uniform distribution with $n$ values is $\frac{1}{n}$ .
What is the sum of probabilities for all values of a discrete random variable?
The sum of probabilities for all values of a discrete random variable is 1.
True or False?
$P (X > k) = 1 - P (X \leq k)$
True.
$P (X > k) = 1 - P (X \leq k)$
True or False?
The probability that random variable $X$ is at least equal to $k$ can be written as $P (X \leq k)$ .
False.
The probability that random variable $X$ is at least equal to $k$ is written as $P (X \geq k)$ .
$P (X \leq k)$ is the probability that $X$ is at most equal to $k$ , or is less than or equal to $k$ .
True or False?
A discrete random variable can only take a finite number of values.
False.
A discrete random variable can take a finite or infinite number of values.
Define expected value.
Expected value is the average outcome of an experiment if it is repeated many times.
It is calculated as the sum of each possible outcome multiplied by its probability.
What is $E (X)$ the notation for?
$E (X)$ is the expected value (or the mean) of a random variable $X$ .
True or False?
The expected value of a random variable is always a whole number.
False.
The expected value of a random variable can be any real number, including fractions or decimals.
True or False?
The expected value $E (X)$ must be an obtainable value of $X$ .
False.
The expected value $E (X)$ does not need to be an obtainable value of $X$ .
For example, the expected value when rolling a fair 6-sided dice is 3.5, even though only the numbers 1, 2, 3, 4, 5 and 6 appear on the dice.
What is the formula for calculating $E (X)$ for a discrete random variable?
The formula for calculating $E (X)$ for a discrete random variable is $E (X) = \sum x P (X = x)$
Where:
- $x$ is a value that the random variable $X$ can take
- $P (X = x)$ is the probability that $X$ takes the value $x$
- $\sum$ means to sum up the products $x P (X = x)$ for all possible values $x$
This equation is in the exam formula booklet.
What is a fair game in terms of expected value?
In terms of expected value, a fair game is one where $E (X) = 0$ .
True or False?
When used with reference to the outcome of a game, a positive value for $E (X)$ means the player can expect to make a loss.
False.
When used with reference to the outcome of a game, a positive value for $E (X)$ means the player can expect to make a gain.
A negative value for $E (X)$ means the player can expect to make a loss.
How is the expected gain or loss typically calculated in a game?
The expected gain or loss for a game is typically calculated by subtracting the cost to play the game from the expected value of the prize.
Define symmetrical distribution.
A symmetrical distribution is one where the possible values of $X$ are symmetrical about a middle value, and their probabilities are also symmetrical.
What is special about the mean of a symmetrical distribution?
The mean of a symmetrical distribution is the same as its median.
How do you calculate $E (X^{2})$ for a discrete random variable $X$ .
To calculate $E (X^{2})$ for a discrete random variable $X$ , you use the formula $E (X^{2}) = \sum x^{2} P (X = x)$ .
For each outcome, multiply its probability by its square and then add together.
This formula is not given explicitly in the formula booklet.
True or False?
$E (X^{2}) = {(E (X))}^{2}$ .
False.
In general, $E (X^{2}) \neq {(E (X))}^{2}$ .
True or False?
$Var (X) = {[E (X)]}^{2} - E (X^{2})$ .
False.
$Var (X)$ is not equal to ${[E (X)]}^{2} - E (X^{2})$ .
The correct formula is $Var (X) = E (X^{2}) - {[E (X)]}^{2}$ .
This is given in the exam formula booklet.
$X$ is a random variable.
Write an expression for $E (a X + b)$ in terms of $E (X)$ , where $a$ and $b$ are constants.
If $X$ is a random variable, then $E (a X + b) = a E (X) + b$ , where $a$ and $b$ are constants.
$X$ is a random variable.
Write an expression for $Var (a X + b)$ in terms of $Var (X)$ , where $a$ and $b$ are constants.
If $X$ is a random variable, then $Var (a X + b) = a^{2} Var (X)$ , where $a$ and $b$ are constants.
Adding or subtracting a constant does not affect the variance.
True or False?
$Var (3 - X) = - Var (X)$ .
False.
Variance can never be negative.
$Var (3 - X) = {(- 1)}^{2} Var (X) = Var (X)$ .

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