The following table shows the probability distribution for the random variable  V*V*.

v*v*

22

33

44

55

66

P(V=v)*P*(*V*=*v*)

0.010.01

c2*c*2

c+0.1*c*+0.1

2c+0.092*c*+0.09

0.160.16

Find the value of the constant  c*c*  and find  P(V>4)*P*(*V*>4).

To solve for the constant \( c \) and find \( P(V>4) \), let's start by identifying the correct probability values from the table. Given:

\[
\begin{array}{c|c}
V & P(V=v) \\
\hline
2 & 0.01 \\
3 & c \\
4 & c + 0.1 \\
5 & 2c + 0.09 \\
6 & 0.16 \\
\end{array}
\]

The sum of the probabilities must equal 1:

\[
0.01 + c + (c + 0.1) + (2c + 0.09) + 0.16 = 1
\]

Combine like terms:

\[
0.01 + c + c + 0.1 + 2c + 0.09 + 0.16 = 1
\]
\[
0.01 + 4c + 0.35 = 1
\]

Simplify the equation:

\[
4c + 0.36 = 1
\]

Isolate \( c \):

\[
4c = 1 - 0.36
\]
\[
4c = 0.64
\]
\[
c = \frac{0.64}{4}
\]
\[
c = 0.16
\]

Now substitute \( c = 0.16 \) back into the probabilities:

\[
P(V=2) = 0.01
\]
\[
P(V=3) = c = 0.16
\]
\[
P(V=4) = c + 0.1 = 0.16 + 0.1 = 0.26
\]
\[
P(V=5) = 2c + 0.09 = 2(0.16) + 0.09 = 0.32 + 0.09 = 0.41
\]
\[
P(V=6) = 0.16
\]

Check the sum:

\[
0.01 + 0.16 + 0.26 + 0.41 + 0.16 = 1
\]

Now, we need to find \( P(V > 4) \):

\[
P(V > 4) = P(V=5) + P(V=6)
\]
\[
P(V > 4) = 0.41 + 0.16 = 0.57
\]

So, the value of the constant \( c \) is \( 0.16 \) and \( P(V > 4) = 0.57 \).

Would you like more details or have any questions?

Here are 5 additional questions you might ask:
1. How do you find the expected value of a discrete random variable?
2. What is the variance and how is it calculated for a probability distribution?
3. Can you explain how to use the cumulative distribution function (CDF)?
4. How do you interpret the standard deviation in a probability distribution?
5. What are some common probability distributions and their applications?

**Tip:** Always verify that your probability distribution sums to 1 before using it for further calculations.

Learn how to find the constant c and probability P(V > 4) using a probability distribution table. This problem involves calculating probabilities for a discrete random variable and solving for a constant in the context of probability distributions. Suitable for college-level students and those interested in probability theory.

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