The problem asks us to find the probability $P(X\geq 3)$ for a discrete random variable X, whose probability mass function (PMF) is given by $P(X= x) = k(x + 1)3^{-x}$ for $x = 0, 1, 2, 3\dots$. Here, $k$ is a constant that needs to be determined first.

Determining the Constant k

The sum of probabilities for all possible values of a random variable must equal 1. Therefore, we have:

$ \sum_{x=0}^{\infty} P(X=x) = 1 $

Substituting the given PMF:

$ \sum_{x=0}^{\infty} k(x + 1)3^{-x} = 1 $

We can factor out the constant $k$:

$ k \sum_{x=0}^{\infty} (x + 1)\left(\frac{1}{3}\right)^x = 1 $

The sum $\sum_{x=0}^{\infty} (x + 1)r^x$ is known to be equal to $\frac{1}{(1-r)^2}$ for $|r|<1$. In this case, $r = 1/3$.

So, the sum evaluates to:

$ \sum_{x=0}^{\infty} (x + 1)\left(\frac{1}{3}\right)^x = \frac{1}{\left(1 - \frac{1}{3}\right)^2} = \frac{1}{\left(\frac{2}{3}\right)^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4} $

Now, substitute this back into the equation for $k$:

$ k \left(\frac{9}{4}\right) = 1 $

Solving for $k$:

$ k = \frac{4}{9} $

The PMF is therefore $P(X=x) = \frac{4}{9}(x + 1)3^{-x}$.

Calculating P(X >= 3)

We want to find $P(X\geq 3)$. It is often easier to calculate this using the complement rule: $P(X\geq 3) = 1 - P(X < 3)$.

$P(X < 3)$ is the sum of probabilities for $x=0, 1, 2$:

$ P(X < 3) = P(X=0) + P(X=1) + P(X=2) $

Let's calculate each term:

• $P(X=0) = \frac{4}{9}(0 + 1)3^{-0} = \frac{4}{9}(1)(1) = \frac{4}{9}$
• $P(X=1) = \frac{4}{9}(1 + 1)3^{-1} = \frac{4}{9}(2)\left(\frac{1}{3}\right) = \frac{8}{27}$
• $P(X=2) = \frac{4}{9}(2 + 1)3^{-2} = \frac{4}{9}(3)\left(\frac{1}{9}\right) = \frac{12}{81} = \frac{4}{27}$

Now, sum these probabilities:

$ P(X < 3) = \frac{4}{9} + \frac{8}{27} + \frac{4}{27} $

To add these fractions, find a common denominator, which is 27:

$ P(X < 3) = \frac{4 \times 3}{9 \times 3} + \frac{8}{27} + \frac{4}{27} = \frac{12}{27} + \frac{8}{27} + \frac{4}{27} = \frac{12 + 8 + 4}{27} = \frac{24}{27} $

Finally, calculate $P(X\geq 3)$:

$ P(X\geq 3) = 1 - P(X < 3) = 1 - \frac{24}{27} = \frac{27 - 24}{27} = \frac{3}{27} $

Simplify the fraction:

$ P(X\geq 3) = \frac{1}{9} $