The problem provides a discrete probability distribution and asks to calculate the expected value $E(X)$ (denoted as $M(x)$ in the image), the mode $M_o$, the variance $D(X)$, and the standard deviation $\sigma(X)$. The given data are: $X = \{0, 15, 20\}$ $P(X) = \{0.1, 0.7, 0.2\}$

1. Problem Description

The problem provides a discrete probability distribution and asks to calculate the expected value $E(X)$ (denoted as $M(x)$ in the image), the mode $M_o$, the variance $D(X)$, and the standard deviation $\sigma(X)$. The given data are:

$X = \{0, 15, 20\}$

$P(X) = \{0.1, 0.7, 0.2\}$

2. Solution Steps

First, we compute the expected value $E(X)$ using the formula:

$E(X) = \sum x_i P(x_i)$

$E(X) = (0 \times 0.1) + (15 \times 0.7) + (20 \times 0.2) = 0 + 10.5 + 4 = 14.5$

Next, we find the mode $M_o$, which is the value of $X$ with the highest probability. In this case, the highest probability is 0.7, which corresponds to $X = 15$. So, $M_o = 15$.

Then, we calculate the variance $D(X)$ using the formula:

$D(X) = E(X^2) - [E(X)]^2$

We already have $E(X) = 14.5$. Now, we need to calculate $E(X^2)$:

$E(X^2) = \sum x_i^2 P(x_i)$

$E(X^2) = (0^2 \times 0.1) + (15^2 \times 0.7) + (20^2 \times 0.2) = (0 \times 0.1) + (225 \times 0.7) + (400 \times 0.2) = 0 + 157.5 + 80 = 237.5$

Now we can compute $D(X)$:

$D(X) = 237.5 - (14.5)^2 = 237.5 - 210.25 = 27.25$

Finally, we calculate the standard deviation $\sigma(X)$ using the formula:

$\sigma(X) = \sqrt{D(X)}$

$\sigma(X) = \sqrt{27.25} \approx 5.22$

3. Final Answer

$E(X) = 14.5$

$M_o = 15$

$D(X) = 27.25$

$\sigma(X) \approx 5.22$