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8.01 Probability density functions

Worksheet
Probability density functions
1

State whether the following graphs are probability density functions.

a
1
2
3
4
5
6
7
x
\frac{1}{6}
f(x)
b
4
8
12
x
\frac{1}{10}
f(x)
c
2
4
6
8
x
\frac{1}{8}
f(x)
d
5
10
15
20
x
\frac{1}{10}
f(x)
e
3
6
9
x
\frac{1}{9}
f(x)
2

State whether the following functions describe continuous probability distributions:

a

f \left( x \right)=0.2 for the interval 1 \leq x \leq 6.

b

f \left( x \right)=\dfrac{x}{12} in the domain \left[0,6 \right]

c

f \left( x \right) = \begin{cases} \dfrac{x^3}{324} & \text{for } 0 \leq x \leq 3 \\ 0 & \text{for all other }x \end{cases}

d

f \left( x \right)=\dfrac{x^2}{21} in the domain 1 \leq x \leq 4

e

f \left( x \right) = \begin{cases} \dfrac{x}{8} & \text{for } 1 \leq x \leq 8 \\ 0 & \text{for all other }x \end{cases}

3

Consider the probability density function p \left( x \right) drawn for a random variable X:

a

Find the area between p \left( x \right) and the x axis.

b

State the features of p \left( x \right) which are also features of all probability distribution functions.

c

Calculate P \left(X \leq 54 \right).

d

Calculate P \left(X \gt 34 \right).

e

Calculate P \left(44 \lt X \leq 53 \right).

f

Calculate P \left(X \leq 56 | X \geq 44 \right) .

10
20
30
40
50
60
70
80
90
x
\frac{1}{70}
p(x)
4

Consider the probability density function p \left( x \right) = \dfrac{1}{40} for 60 \leq x \leq 100 and p \left( x \right) = 0 otherwise.

a

Sketch the graph of the probability density function p \left(x \right).

b

Calculate P \left( X \gt 80 \right).

c

Calculate P \left(X \leq 65 | X \leq 90 \right).

d

Calculate P \left(X \leq 70 \cap X \geq 80 \right).

5

Consider the probability density function p \left( x \right) drawn for a random variable X:

a

Find the area between p \left( x \right) and the \\ x-axis.

b

State features of p \left( x \right) which are also features of all continuous probability distribution functions.

c

Calculate P \left(X \lt 3 \right).

d

Calculate P \left(X \gt 3 | X \leq 4 \right).

1
2
3
4
5
6
7
8
x
\frac{2}{25}
\frac{4}{25}
\frac{6}{25}
\frac{8}{25}
\frac{2}{5}
p(x)
6

Consider the probability density function p \left( x \right) drawn for a random variable X:

a

Find the area between p \left( x \right) and the x-axis.

b

State the features of p \left( x \right) which are also features of all continuous probability distribution functions.

c

Calculate P \left (\dfrac{11}{2} \leq X \lt \dfrac{17}{2} \right).

d

Calculate P \left (X \lt \dfrac{17}{2}|X \geq \dfrac{11}{2} \right).

1
2
3
4
5
6
7
8
9
10
11
x
\frac{1}{12}
\frac{1}{6}
\frac{1}{4}
p(x)
7

Consider the probability density function p \left( x \right) drawn for a random variable X:

a

Find the area between p \left( x \right) and the \\ x-axis.

b

State the features of p \left( x \right) which are also features of all continuous probability distribution functions.

c

State the probability density function for X.

d

Calculate P \left(X \lt 2|X \gt 1 \right).

2
4
6
8
10
12
14
x
\frac{1}{10}
p(x)
8

Consider the probability density function p \left( x \right) drawn for a random variable X:

a

Find the area between p \left( x \right) and the \\ x-axis.

b

State the features of p \left( x \right) which are also features of all probability distribution functions.

c

State the probability density function for X.

d

Calculate P \left(X \geq 4 \right).

e

Calculate P \left(X \gt 3|X \lt 4 \right).

1
2
3
4
5
6
7
8
x
\frac{1}{3}
p(x)
9

Consider the probability density function:

p\left( x \right) = \begin{cases} \dfrac{-x}{40}+\dfrac{1}{5}, & 0 \leq x \leq 8 \\ \dfrac{x}{160}-\dfrac{1}{20}, & 8 \lt x \leq 16 \\ 0, & \text{otherwise} \end{cases}

a

Sketch the graph of the probability density function p\left( x \right).

b

Calculate the area under p \left( x \right).

c

Explain the reason why p \left( x \right) is a probability density function.

10

The probability density function of a random variable X is shown. Its non-zero values lie in the region 0 \leq x \leq k:

a

Calculate the value of k.

b

State the equation that defines the probability distribution function of X in the domain 0 \leq x \leq k.

c

Calculate P \left(X \lt 0.8 k \right) to two decimal places.

d

Calculate P \left(X \geq 0.2 k \right) to two decimal places.

e

Calculate P \left(X \lt 7 | X \gt 2 \right).

x
\frac{1}{10}
\frac{1}{5}
\frac{3}{10}
p(x)
11

Let f \left(x \right)=\dfrac{3}{4} \left(x^2-4x+3 \right) be a function defined on the closed interval \left[0,4 \right].

a

Calculate \int_0^4 f \left(x \right)dx.

b

Sketch the graph of y=f \left(x \right).

c

State whether f \left(x \right) is a PDF.

12

A probability density function has the equation f \left(x \right)=\dfrac{x^4}{3355} over the domain \left[2,b \right].

Find the value of b.

13

Consider the continuous probability distribution f \left(x \right)=\begin{cases} kx^3 & \text{for } 0 \leq x \leq 5 \\ 0 & \text{for all other } x \end{cases}.

Find the value of k.

14

A function is given by f \left(x \right)=\dfrac{x^2}{72}. Find the domain of this probability density function starting at x=0.

15

A PDF is given by f \left(x \right)=\dfrac{2x^5}{87\,381} over the interval 1 \leq x \leq b. Find the value of b.

16

The continuous probability distribution is defined by f \left(x \right)=ax^2 on the interval 0 \leq x \leq 5.

a

Find the value of a.

b

Calculate:

i
P \left(X \leq 3 \right)
ii
P \left(1 \lt X \lt 4 \right)
iii
P \left(X \gt 2 \right)
iv
P \left(X \lt 1 \right)
v
P \left(3 \leq X \lt 4 \right)
17

The continuous random variable X has the PDF shown:

a

Find the value of a.

b

Calculate:

i
P \left(1 \leq X \leq 3 \right)
ii
P \left(X \lt 2 \right)
iii
P \left(1 \leq X \leq 2 \right)
iv
P \left(X \leq 1 \right)
1
2
3
x
0.4
f(x)
18

A continuous probability function is given by f \left(x \right)=ke^x, defined on the domain \left[1,6 \right].

a

Find the exact value of k.

b

Calculate:

i
P \left(2 \leq X \leq 5 \right)
ii
P \left(X \lt 4 \right)
iii
P \left(X \geq 3 \right)
19

Consider y=\sin x.

a

Is y=\sin x a PDF on the domain \left[0, \dfrac{\pi}{2}\right]?

b

Calculate:

i
P\left(X \leq \dfrac{\pi}{3}\right)
ii
P\left(0 \lt X \lt \dfrac{\pi}{4}\right)
iii
P\left(X \gt \dfrac{\pi}{6}\right)
20

The mass of a 6 week old puppy, in grams, is modeled by a continuous random variable X which has probability density function p \left( x \right) defined by \\ p \left( x \right)=\begin{cases} k \sin \left( \dfrac{\pi}{240} \left(x - 450\right)\right) & \text{for } 450 \leq x \leq 690 \\ 0 & \text{otherwise} \end{cases}

a

Find the value of k.

b

Calculate the probability that a randomly chosen puppy weighs less than 550 grams.

c

Calculate the probability that a randomly chosen 6 week puppy weighs more than 570 grams, if we are told that it weighs less than 630 grams.

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MA12-8

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