Let's break down the problem and solve each part step-by-step.

Given Function:

The probability mass function (PMF) is provided as: $f(x) = \frac{4x + 3}{55}, \quad x = 0, 1, 2, 3, 4$

To verify that it is a probability mass function (PMF), we need to check two conditions:

1. Each probability $f(x) \geq 0$ for all values of $x$.
2. The sum of the probabilities over all possible values of $x$ should equal 1: $\sum_{x=0}^{4} f(x) = 1$

Step 1: Check if the function is a valid PMF

We'll first calculate the values of $f(x)$ for each $x$.

For $x = 0$: $f(0) = \frac{4(0) + 3}{55} = \frac{3}{55}$

For $x = 1$: $f(1) = \frac{4(1) + 3}{55} = \frac{7}{55}$

For $x = 2$: $f(2) = \frac{4(2) + 3}{55} = \frac{11}{55}$

For $x = 3$: $f(3) = \frac{4(3) + 3}{55} = \frac{15}{55}$

For $x = 4$: $f(4) = \frac{4(4) + 3}{55} = \frac{19}{55}$

Now, sum these values: $\frac{3}{55} + \frac{7}{55} + \frac{11}{55} + \frac{15}{55} + \frac{19}{55} = \frac{55}{55} = 1$

Since the sum is 1, the function satisfies the conditions of a probability mass function.

Step 2: Solve the given probability questions

(a) $P(X = 4)$

From the calculation above: $P(X = 4) = f(4) = \frac{19}{55}$

(b) $P(X \leq 1)$

This is the probability that $X$ is 0 or 1: $P(X \leq 1) = f(0) + f(1) = \frac{3}{55} + \frac{7}{55} = \frac{10}{55}$

(c) $P(2 \leq X < 4)$

This is the probability that $X$ is 2 or 3: $P(2 \leq X < 4) = f(2) + f(3) = \frac{11}{55} + \frac{15}{55} = \frac{26}{55}$

(d) $P(X > -10)$

Since $X$ can only take the values $0, 1, 2, 3, 4$, and all of these are greater than -10, the probability is 1: $P(X > -10) = 1$

Final Answers:

• (a) $P(X = 4) = \frac{19}{55}$
• (b) $P(X \leq 1) = \frac{10}{55}$
• (c) $P(2 \leq X < 4) = \frac{26}{55}$
• (d) $P(X > -10) = 1$

Would you like further details on any of these steps?

5 Related Questions:

1. What is the general definition of a probability mass function?
2. How would the PMF change if the probabilities didn't sum to 1?
3. What is the difference between a PMF and a probability density function (PDF)?
4. How can you calculate the cumulative distribution function (CDF) from a PMF?
5. Can PMFs have negative values for any $f(x)$?

Tip:

Always verify that the sum of probabilities equals 1 when dealing with discrete distributions like PMFs to ensure it's valid.