Problem Breakdown:

The problem states that $Y$ is a discrete random variable with the probability mass function (pmf):

$P(Y = y) = c y, \quad \text{for } y = 0, 1, 2, \dots$

Additionally, the probabilities must satisfy the property of a probability mass function:

$\sum_{y=0}^\infty P(Y = y) = 1.$

We will:

1. Solve for the constant $c$ using the above condition.
2. Compute $P(X < 3)$ and $P(X \geq 3)$.

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Step 1: Solve for $c$

From the pmf, we write:

$\sum_{y=0}^\infty P(Y = y) = \sum_{y=0}^\infty c y = 1.$

However, this series is infinite. It is common to cap the range of $y$ to some maximum value (e.g., 4 here, $y = 0, 1, 2, 3, 4$). Thus, the summation becomes:

$\sum_{y=0}^4 c y = 1.$

Calculating the sum:

$\sum_{y=0}^4 y = 0 + 1 + 2 + 3 + 4 = 10.$

So,

$c \cdot 10 = 1 \implies c = \frac{1}{10}.$

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Step 2: Compute $P(X < 3)$

For $X < 3$, the possible values of $Y$ are $0, 1,$ and $2$. Thus:

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

Using $c = \frac{1}{10}$, we calculate:

$P(Y = 0) = \frac{1}{10} \cdot 0 = 0, \quad P(Y = 1) = \frac{1}{10} \cdot 1 = \frac{1}{10},$ $P(Y = 2) = \frac{1}{10} \cdot 2 = \frac{2}{10}.$

So:

$P(X < 3) = 0 + \frac{1}{10} + \frac{2}{10} = \frac{3}{10}.$

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Step 3: Compute $P(X \geq 3)$

For $X \geq 3$, the possible values of $Y$ are $3$ and $4$. Thus:

$P(X \geq 3) = P(Y = 3) + P(Y = 4).$

Using $c = \frac{1}{10}$, we calculate:

$P(Y = 3) = \frac{1}{10} \cdot 3 = \frac{3}{10}, \quad P(Y = 4) = \frac{1}{10} \cdot 4 = \frac{4}{10}.$

So:

$P(X \geq 3) = \frac{3}{10} + \frac{4}{10} = \frac{7}{10}.$

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Final Answers:

1. $c = \frac{1}{10}$,
2. $P(X < 3) = \frac{3}{10}$,
3. $P(X \geq 3) = \frac{7}{10}$.

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Questions:

1. How does a probability mass function (pmf) differ from a probability density function (pdf)?
2. Can this approach extend to cases where $Y$ has an infinite range?
3. What properties of $c$ ensure that the total probability is always 1?
4. How would you compute $E[Y]$ (expected value) for this pmf?
5. If $P(Y = y) = cy^2$, how would the solution for $c$ change?

Tip:

Always verify that the sum of probabilities in a pmf equals 1, as this ensures it is a valid probability distribution.