Question

Which of the following tables shows a valid probability density function? a. x P(X=x) 0 38 1 14 2 38 b. x P(X=x) 0 0.2 1 0.1 2 0.35 3 0.17 c. x P(X=x) 0 910 1 −310 2 310 3 110 d. x P(X=x) 0 0.06 1 0.01 2 0.07 3 0.86 e. x P(X=x) 0 12 1 18 2 14 3 18 f. x P(X=x) 0 110 1 110 2 310 3

Answer 1

Explanation:

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

a. 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

b. 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

c. 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

d. 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

e. 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

f. 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