Question

Which of the following is a valid probability distribution?Probability Distribution AProbability Distribution BProbability Distribution CProbability Distribution D

Answer 1

If a distribution satisfies the following cases

Case 1

$0\leq p(x)\leq1$

Case 2

$\sum ^N_{n\mathop=1}p(x_n)=1$

Then the distribution is called a probability distribution.

In distribution A

$p(x)=0.2$

For each x but

$\sum ^{}_{}p(x)=0.2+0.2+0.2+0.2+0.2+0.2=1.2$

Hence it is not a probability distribution.

In B

$\sum ^{}_{}p(x)=0.1+0.2+0.3+0.3+0.2+0.1=1.2\ne1$

Hence this is not a probability distribution.

In the distribution C

$\sum ^{}_{}p(x)=0.1+0.2+0.4+0.1+0.1=0.9\ne1_{}$

Hence this is not a probability distribution.

In case of D

$\sum ^{}_{}p(x)=0.2+0.1+0.1+0.5+0+0.1=1$

And for all values of x

$0\leq p(x)\leq1$

Hence D is a valid probability distribution.