New Zealand
Level 8 - NCEA Level 3

# Discrete Probability Density Functions

Lesson

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:

## Properties of the distribution of a DRV

It's so important to remember these three properties, that we'll put them here again:

1. All outcomes are numerical and discrete - this means we can't have categorical or continuous numerical data.
2. The probabilities of all possible outcomes add to $1$1 - this makes sense when you think about it, if they didn't all add up to $1$1 then we must be missing one of the outcomes or have one of the probabilities wrong.
3. The probability $p$p of all outcomes is greater than or equal to $0$0 and less than or equal to $1$1, that is $0\le p\le1$0p1.

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.

#### Worked Examples

##### Question 1

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
1. Determine the value of $k$k.

2. Calculate $P$P$($($X<3$X<3$)$).

3. Calculate $P$P$($($X\ge2$X2 $|$| $X<4$X<4$)$).

4. Determine $m$m such that $P$P$($($X\ge m$Xm$)=0.75$)=0.75.

##### Question 2

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
1. Determine the value of $k$k.

2. Hence complete the table of values.

 $x$x $P$P$($($X=x$X=x$)$) $4$4 $5$5 $6$6 $7$7 $8$8 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. Calculate $P$P$($($X<7$X<7$)$).

4. Calculate $P$P$($($X\ge6$X6$)$).

5. Calculate $P$P$($($X<6$X<6$\cup$$X>7$X>7$)$).

6. Calculate $P$P$($($X\ge5$X5$|$|$X\le7$X7$)$).

##### Question 3

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
1. Complete the table of values.

Give each value to five decimal places.

 $x$x $P(X=x)$P(X=x) $0$0 $1$1 $2$2 $3$3 $4$4 $5$5 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Which of the following describes the shape of the probability distribution?

negatively skewed

A

positively skewed

B

symmetrical

C

negatively skewed

A

positively skewed

B

symmetrical

C
3. Calculate $P$P$($($X>3$X>3$)$).

4. Calculate $P$P$($($X>0$X>0$)$).

5. Calculate $P$P$($($X\le3$X3$|$|$X>0$X>0$)$).

### Outcomes

#### S8-4

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

#### 91586

Apply probability distributions in solving problems