Question

Which of the following tables shows a valid probability density function? Select all correct answers.

Select all that apply:

• x	P(X=x)
  0	3/8
  1	1/4
  2	3/8

• x	P(X=x)
  0	0.2
  1	0.1
  2	0.35
  3	0.17

• x	P(X=x)
  0	9/10
  1	−3/10
  2	3/10
  3	1/10

• x	P(X=x)
  0	0.06
  1	0.01
  2	0.07
  3	0.86

• x	P(X=x)
  0	1/2
  1	1/8
  2	1/4
  3	1/8

• x	P(X=x)
  0	1/10
  1	1/10
  2	3/10
  3	1/5

Answer 1

Since we know that for a distribution be a probability density function sum of all the probability events should be equal to 1 and all individual events should have probability between 0 and 1

x	P(X=x)
0	3/8
1	1/4
2	3/8

P(X=0)+P(X=1)+P(X=2) = 3/8 + 1/4 + 3/8

P(X=0)+P(X=1)+P(X=2) = 6/8 + 2/8 = 1

This is a probability density function

x	P(X=x)
0	0.2
1	0.1
2	0.35
3	0.17

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 0.2 + 0.1 + 0.35 + 0.17

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 0.65 + 0.17 = 0.82 1

Therefore this is NOT a probability density function

x	P(X=x)
0	9/10
1	−3/10
2	3/10
3	1/10

Since P(X=1) is not between 0 and 1

Therefore this is NOT a probability density function

x	P(X=x)
0	0.06
1	0.01
2	0.07
3	0.86

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 0.06 + 0.01 + 0.07 + 0.86

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 0.14 + 0.86 = 1

Therefore this is a probability density function

x	P(X=x)
0	1/2
1	1/8
2	1/4
3	1/8

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 1/2 + 1/8 + 1/4 + 1/8

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 1/2 + 1/2 = 1

Therefore this is a probability density function

x	P(X=x)
0	1/10
1	1/10
2	3/10
3	1/5

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 1/10 + 1/10 + 3/10 + 1/5

P(X=0)+P(X=1)+P(X=2)+P(X=3) = 2/10 + 5/10 = 7/10 1

Therefore this is NOT a probability density function