As we've seen, we can represent a probability distribution for a discrete random variable (DRV) with a table of probabilities or a graph.
We can sometimes also represent a probability distribution for a DRV with what we call a probability density function or a probability distribution function. In other words, we can write an algebraic equation that will generate the probability for each possible outcome.
For example, if we consider the simple example of all the possible outcomes when we roll a normal six-sided dice, we would obtain the following probability distribution function:
It's so important to remember these three properties, that we'll put them here again:
If you are required to write your own probability density function from some information given, then make sure you remember to write all the possible domain values (that is, all the possible outcomes) for which the function conforms to the properties stated above.
The probability function for a uniform discrete random variable is given below:
$P$P$($($X=x$X=x$)$) | $=$= | $k$k; $x=1,2,3,4$x=1,2,3,4 | |
$0$0, for all other values of $x$x |
Determine the value of $k$k.
Calculate $P$P$($($X<3$X<3$)$).
Calculate $P$P$($($X\ge2$X≥2 $|$| $X<4$X<4$)$).
Determine $m$m such that $P$P$($($X\ge m$X≥m$)=0.75$)=0.75.
The probability function for a discrete random variable is given by:
$P$P$($($X=x$X=x$)$) | $=$= | $k\left(9-x\right)$k(9−x); $x=4,5,6,7,8$x=4,5,6,7,8 | |
$0$0, for all other values of $x$x |
Determine the value of $k$k.
Hence complete the table of values.
$x$x | $4$4 | $5$5 | $6$6 | $7$7 | $8$8 |
---|---|---|---|---|---|
$P$P$($($X=x$X=x$)$) | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Calculate $P$P$($($X<7$X<7$)$).
Calculate $P$P$($($X\ge6$X≥6$)$).
Calculate $P$P$($($X<6$X<6$\cup$∪$X>7$X>7$)$).
Calculate $P$P$($($X\ge5$X≥5$|$|$X\le7$X≤7$)$).
The probability function for a discrete random variable is given by:
$P$P$($($X=x$X=x$)$) | $=$= | $\nCr{5}{x}$5Cx$\left(0.6\right)^x$(0.6)x$\left(0.4\right)^{5-x}$(0.4)5−x; $x=0,1,2,3,4,5$x=0,1,2,3,4,5 | |
$0$0, for all other values of $x$x |
Complete the table of values.
Give each value to five decimal places.
$x$x | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
---|---|---|---|---|---|---|
$P(X=x)$P(X=x) | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Which of the following describes the shape of the probability distribution?
negatively skewed
positively skewed
symmetrical
negatively skewed
positively skewed
symmetrical
Calculate $P$P$($($X>3$X>3$)$).
Give your answer to five decimal places.
Calculate $P$P$($($X>0$X>0$)$).
Give your answer to five decimal places.
Calculate $P$P$($($X\le3$X≤3$|$|$X>0$X>0$)$).
Give your answer to five decimal places.
Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal
Apply probability distributions in solving problems