The discrete random variable  R*R*  is such that  R∈{1,2,4,5}*R*∈{1,2,4,5}.Given that  P(R=r)=k(r+1)r+2*P*(*R*=*r*)=*r*+2*k*(*r*+1)​, then find.

**(a)**

The value of the constant  k*k*.

**(b)**

P(R⩽3)*P*(*R*⩽3).

To solve the given problem, we'll use the same approach as before. The discrete random variable \( R \) takes on the values \(\{1, 2, 4, 5\}\) with probabilities defined by \( P(R = r) = \frac{k(r+1)}{r+2} \).

### Part (a) Find the value of the constant \( k \).

Since \( R \) is a discrete random variable, the sum of all probabilities must equal 1:

\[
\sum_{r \in \{1, 2, 4, 5\}} P(R = r) = 1
\]

Let's compute each probability:

\[
P(R = 1) = \frac{k(1+1)}{1+2} = \frac{2k}{3}
\]

\[
P(R = 2) = \frac{k(2+1)}{2+2} = \frac{3k}{4}
\]

\[
P(R = 4) = \frac{k(4+1)}{4+2} = \frac{5k}{6}
\]

\[
P(R = 5) = \frac{k(5+1)}{5+2} = \frac{6k}{7}
\]

Now, sum these probabilities and set them equal to 1:

\[
\frac{2k}{3} + \frac{3k}{4} + \frac{5k}{6} + \frac{6k}{7} = 1
\]

To combine these fractions, find a common denominator. The least common multiple of 3, 4, 6, and 7 is 84:

\[
\frac{2k \cdot 28}{84} + \frac{3k \cdot 21}{84} + \frac{5k \cdot 14}{84} + \frac{6k \cdot 12}{84} = 1
\]

\[
\frac{56k}{84} + \frac{63k}{84} + \frac{70k}{84} + \frac{72k}{84} = 1
\]

Combine the numerators:

\[
\frac{56k + 63k + 70k + 72k}{84} = 1
\]

\[
\frac{261k}{84} = 1
\]

Solving for \( k \):

\[
k = \frac{84}{261} = \frac{4}{13}
\]

### Part (b) Find \( P(R \leq 3) \).

Since \( R \) can only be \( 1 \) or \( 2 \) and both are \(\leq 3\):

\[
P(R \leq 3) = P(R = 1) + P(R = 2)
\]

Substitute the value of \( k \):

\[
P(R = 1) = \frac{2k}{3} = \frac{2 \cdot \frac{4}{13}}{3} = \frac{8}{39}
\]

\[
P(R = 2) = \frac{3k}{4} = \frac{3 \cdot \frac{4}{13}}{4} = \frac{12}{52} = \frac{3}{13}
\]

Adding these probabilities:

\[
P(R \leq 3) = \frac{8}{39} + \frac{3}{13}
\]

To combine these fractions, find a common denominator. The least common multiple of 39 and 13 is 39:

\[
\frac{8}{39} + \frac{3 \cdot 3}{13 \cdot 3} = \frac{8}{39} + \frac{9}{39} = \frac{17}{39}
\]

So, the answers are:
- (a) \( k = \frac{4}{13} \)
- (b) \( P(R \leq 3) = \frac{17}{39} \)

Would you like further details on any part of this solution or have any questions?

Here are some additional questions you might find useful:
1. How do you find the least common multiple of denominators?
2. What is the significance of the sum of probabilities equaling 1 in a probability distribution?
3. How do you convert a probability fraction into a decimal?
4. Can you explain more about discrete random variables and their properties?
5. How can we check if a probability distribution is valid?

**Tip:** Always double-check your calculations, especially when finding a common denominator and combining fractions.

Explore how to find the constant k and the probability P(R ≤ 3) for the discrete random variable R, where R takes values from {1, 2, 4, 5} with specific probabilities. Understand the calculations step-by-step and learn about discrete random variables and probability distributions. Suitable for college-level students and learners.

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