Question

Find the constant c in the probability mass function p(x)=(c/(x+1)(x+2)), x=0,1,2,3, ...

Answer 1

The constant
`\( c \) in the probability mass function is \( (2)/(3) \).`

Step-by-step explanation:

The given probability mass function is
`\( p(x) = (c)/((x+1)(x+2)) \) for \( x = 0, 1, 2, 3, \ldots \).` To find the constant
`\( c \)`, we can use the fact that the sum of probabilities in a probability mass function must equal 1. In other words,
`\( \sum_(x=0)^(\infty) p(x) = 1 \).`

Substituting the given function into the sum, we get
`\( \sum_(x=0)^(\infty) (c)/((x+1)(x+2)) \).` We can simplify this expression by partial fraction decomposition. The decomposition leads to
`\( (c)/(2) \left((1)/(x+1) - (1)/(x+2)\right) \).`Now, when we sum this expression, most terms cancel out, leaving
`\( (c)/(2) \cdot (1)/(1+2) \)` as the remaining term.

Setting this equal to 1 (since the sum of probabilities must equal 1), we solve for
`\( c \) and find \( c = (2)/(3) \).` Therefore, the constant
`\( c \) in the probability mass function is \( (2)/(3) \),` ensuring the probabilities sum to 1 over all possible values of \( x \).