Question

let (x) be a random variable whose pdf is given by the plot below. assume that the pdf is zero outside of the interval given in the plot.
estimate the mean of (x).
estimate the standard deviation of (x) to within 0.5.
for which (a) is (p(aleq xleq a + 2)) maximized?
estimate (p(0leq xleq 2)) to two decimal places.
submit question

Explanation:

Step1: Estimate the mean

The mean of a symmetric distribution (the given pdf looks approximately symmetric) is at the center of symmetry. From the plot, the center of the distribution seems to be around $x = 2$. So, we estimate the mean $\mu\approx2$.

Step2: Estimate the standard - deviation

For a normal - like distribution, about 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Looking at the plot, the interval that contains about 95% of the data seems to be from approximately $x = 0$ to $x = 4$. If we assume this is a two - standard - deviation interval ($\mu - 2\sigma$ to $\mu+2\sigma$) and $\mu = 2$, then $4=2 + 2\sigma$ and $0=2-2\sigma$, which gives $\sigma\approx1$.

Step3: Maximize $P(a\leq X\leq a + 2)$

The probability $P(a\leq X\leq a + 2)$ is maximized when the interval $[a,a + 2]$ contains the most of the area under the pdf curve. Since the distribution is symmetric about $x = 2$, when $a=1$, the interval $[1,3]$ is centered around the peak of the distribution. So, $a = 1$ maximizes $P(a\leq X\leq a + 2)$.

Step4: Estimate $P(0\leq X\leq 2)$

We can estimate the area under the curve from $x = 0$ to $x = 2$ by visually comparing it to the total area under the curve (which is 1). The area from $x = 0$ to $x = 2$ seems to be about 0.35.

Answer:

Estimate the mean of $X$: 2
Estimate the standard deviation of $X$ to within 0.5: 1
For which $a$ is $P(a\leq X\leq a + 2)$ maximized: 1
Estimate $P(0\leq X\leq 2)$ to two decimal places: 0.35