For each probability and percentile problem, draw the picture.

Question

Let X ~ Exp(0.2).
Part (g)
Sketch a new graph, shade the area corresponding to the 40th percentile, and find the value. (Round your answer to two decimal places.)

Answer 1

The exponential distribution is a continuous probability distribution with a graph that starts at the origin and decays exponentially to the right. Since the rate parameter is 0.2, the curve is steeper compared to the exponential distribution with a smaller rate parameter.

To find the 40th percentile, we need to find the value of x for which 40% of the distribution lies to the left of x.

To sketch the graph, we start at the origin and draw a curve that decays exponentially to the right. We then shade the area under the curve to the left of the 40th percentile.

Let's say the x-axis represents the possible values of X, and the y-axis represents the probability density function (PDF) of X. The shaded area represents the cumulative distribution function (CDF), which gives the probability that X is less than or equal to a specific value.

The 40th percentile represents the value of x for which the cumulative probability is 0.4.

To find this value, we can use the exponential distribution's cumulative distribution function (CDF) formula:

CDF(x) = 1 - e^(-λx)

where λ = 0.2 (rate parameter) and CDF(x) is the cumulative probability of X being less than or equal to x.

Now we can solve for x:

0.4 = 1 - e^(-0.2x)

Rearranging the equation:

e^(-0.2x) = 1 - 0.4

e^(-0.2x) = 0.6

Taking the natural logarithm (ln) of both sides:

-0.2x = ln(0.6)

Dividing both sides by -0.2:

x ≈ -ln(0.6)/0.2 ≈ 3.48 (rounded to two decimal places)

So the 40th percentile of the exponential distribution with a rate parameter of 0.2 is approximately 3.48.

The final sketch would show the exponential curve, shading the area to the left of x = 3.48, indicating the 40th percentile.