The image you uploaded contains several questions based on a probability distribution model. The distribution is defined as $X \sim \text{Exp}(0.2)$, which represents an exponential distribution with a rate parameter of $\lambda = 0.2$.

Let’s go through the questions one by one.

44. What type of distribution is this?

The distribution is Exponential with a rate parameter $\lambda = 0.2$. This distribution models the time between events in a Poisson process, which is often used in situations where events happen continuously and independently at a constant average rate.

45. Are outcomes equally likely in this distribution? Why or why not?

No, outcomes are not equally likely. The exponential distribution has a memoryless property, meaning the probability of waiting for a certain time depends on the amount of time that has already passed. The distribution is skewed, with more likelihood of smaller values and less likelihood of larger values. The probability density function (PDF) is higher for small values of $x$ and decreases exponentially as $x$ increases.

46. What is m? What does it represent?

In an exponential distribution $X \sim \text{Exp}(\lambda)$, $m$ refers to the mean or expected value of the distribution. For the exponential distribution, the mean is given by $\frac{1}{\lambda}$.

For $\lambda = 0.2$, the mean is $m = \frac{1}{0.2} = 5$.

47. What is the mean?

The mean is the same as $m$ (explained in question 46). The mean is 5.

48. What is the standard deviation?

For an exponential distribution, the standard deviation is also equal to the mean, $\frac{1}{\lambda}$. Therefore, the standard deviation is also 5.

49. State the probability density function.

The probability density function (PDF) for an exponential distribution is given by:

$f(x) = \lambda e^{-\lambda x}, \quad x \geq 0$

Where $\lambda = 0.2$. Therefore, the PDF is:

$f(x) = 0.2 e^{-0.2x}, \quad x \geq 0$

50. Graph the distribution.

The graph of this distribution is an exponentially decaying curve, starting from the maximum value at $x = 0$ and decreasing as $x$ increases.

Would you like me to generate a graph for this distribution?

51. Find $P(2 < x < 10)$.

To calculate $P(2 < x < 10)$, we can use the cumulative distribution function (CDF) of the exponential distribution, which is:

$F(x) = 1 - e^{-\lambda x}$

Thus, to find $P(2 < x < 10)$:

$P(2 < x < 10) = F(10) - F(2)$

Let’s calculate that:

$F(10) = 1 - e^{-0.2 \times 10} = 1 - e^{-2} \approx 1 - 0.1353 = 0.8647$

$F(2) = 1 - e^{-0.2 \times 2} = 1 - e^{-0.4} \approx 1 - 0.6703 = 0.3297$

So:

$P(2 < x < 10) = 0.8647 - 0.3297 = 0.535$

52. Find $P(x > 6)$.

To find $P(x > 6)$, we use the CDF again:

$P(x > 6) = 1 - F(6)$

Where:

$F(6) = 1 - e^{-0.2 \times 6} = 1 - e^{-1.2} \approx 1 - 0.3012 = 0.6988$

So:

$P(x > 6) = 1 - 0.6988 = 0.3012$

Would you like further details or clarification on any of these questions?