True or False? To find the mode of a continuous random variable, you need to integrate its probability density function .

False. To find the mode of a continuous random variable, you do not integrate its probability density function . You can differentiate the probability density function to help find the maximum point .

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

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

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FrontProbability Density Function
True or False?
If $f$ is the probability density function of the continuous random variable $X$ , then $f (x) = P (X = x)$ .

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Cards in this collection (14)

True or False?
If $f$ is the probability density function of the continuous random variable $X$ , then $f (x) = P (X = x)$ .
False.
If $f$ is the probability density function of the continuous random variable $X$ , then $f (x) \neq P (X = x)$ .
The probability density function does not give probabilities of individual values.
If $X$ is a continuous random variable with a probability density function $f$ , how would you find $P (a \leq X \leq b)$ ?
If $X$ is a continuous random variable with a probability density function $f$ , then $P (a \leq X \leq b) = \int_{a}^{b} f (x) d x$ .
True or False?
If $X$ is a continuous random variable, then $P (X = k) = 0$ for any value $k$ .
True.
If $X$ is a continuous random variable, then $P (X = k) = 0$ for any value $k$ .
True or False?
If $X$ is a continuous random variable, then $P (a \leq X \leq b) = P (a < X < b)$ .
True.
If $X$ is a continuous random variable, then $P (a \leq X \leq b) = P (a < X < b)$ .
This is because $P (X = a) = 0$ and $P (X = b) = 0$ .
What are the two conditions a function $f$ needs to meet if it is to be a probability density function?
If $f$ is to be a probability density function then it needs to meet the following two conditions:
- $f$ is never negative, i.e. $f (x) \geq 0$ ,
- the area under the graph of $f$ is equal to 1, i.e. $\int_{- \infty}^{\infty} f (x) d x = 1$ .
True or False?
If $P (X < m) = 0.5$ , then $m$ is the mode of the continuous random variable $X$ .
False.
If $P (X < m) = 0.5$ , then $m$ is the median of the continuous random variable $X$ .
How would you find the median of a continuous random variable $X$ using its probability density density function $f$ ?
To find the median of a continuous random variable, you would integrate the probability density function between its lowest value and the unknown $m$ . Set this definite integral equal to 0.5 and solve for $m$ , which is the median.
I.e. solve the equation $\int_{- \infty}^{m} f (x) d x = 0.5$ for $m$ .
How can you find the mode of a continuous random variable $X$ ?
You can find the mode of a continuous random variable $X$ by finding the value of $x$ which gives the maximum value of the probability density function $f$ .
True or False?
To find the mode of a continuous random variable, you need to integrate its probability density function.
False.
To find the mode of a continuous random variable, you do not integrate its probability density function.
You can differentiate the probability density function to help find the maximum point.
If the probability density function is symmetrical about the line $x = a$ , what is the median of the continuous random variable?
If the probability density function is symmetrical about the line $x = a$ , then the median of the continuous random variable is $x = a$ .
True or False?
If the probability density function is symmetrical about the line $x = a$ , then the mean of the continuous random variable is $a$ .
True.
If the probability density function is symmetrical about the line $x = a$ , then the mean of the continuous random variable is $a$ .
If a probability density function is symmetrical then the mean is the same as the median.
How do you find the mean of a continuous random variable $X$ ?
To find the mean of a continuous random variable $X$ , you multiply the probability density function by $x$ and integrate over its domain.
The formula $E (X) = \int_{- \infty}^{\infty} x f (x) d x$ is given in the exam formula booklet.
How do you find the variance of a continuous random variable $X$ ?
To find the variance of a continuous random variable $X$ , you multiply the probability density function by $x^{2}$ , integrate over its domain and then subtract the mean squared.
The formula $Var (X) = \int_{- \infty}^{\infty} x^{2} f (x) d x - μ^{2}$ is given in the exam formula booklet.
How do you calculate $E (X^{2})$ for the continuous random variable $X$ ?
To calculate $E (X^{2})$ for the continuous random variable $X$ , you use the formula $E (X^{2}) = \int_{- \infty}^{\infty} x^{2} f (x) d x$ .
Multiply the probability density function by $x^{2}$ and integrate it over its domain.
This is not given explicitly in the exam formula booklet.

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