The problem provides the distribution of a discrete random variable $X$ and asks us to find the value of $a$, the distribution function $F(x)$, the mode, the expected value $M(X)$, the variance $D(X)$, and the standard deviation $\sigma(X)$. The distribution is given by: $P(X=1) = 0.3$ $P(X=2) = 0.4$ $P(X=4) = a$ $P(X=8) = 2a$

1. Problem Description

The problem provides the distribution of a discrete random variable $X$ and asks us to find the value of $a$, the distribution function $F(x)$, the mode, the expected value $M(X)$, the variance $D(X)$, and the standard deviation $\sigma(X)$. The distribution is given by:

$P(X=1) = 0.3$

$P(X=2) = 0.4$

$P(X=4) = a$

$P(X=8) = 2a$

2. Solution Steps

1) Find the value of $a$:

Since the sum of probabilities must equal 1, we have:

$0.3 + 0.4 + a + 2a = 1$

$0.7 + 3a = 1$

$3a = 1 - 0.7$

$3a = 0.3$

$a = 0.1$

2) Find the distribution function $F(x)$:

The distribution function $F(x)$ is defined as $F(x) = P(X \le x)$.

$F(x) = \begin{cases}

0, & x < 1 \\

0.3, & 1 \le x < 2 \\

0.3 + 0.4 = 0.7, & 2 \le x < 4 \\

0.7 + a = 0.7 + 0.1 = 0.8, & 4 \le x < 8 \\

0.8 + 2a = 0.8 + 2(0.1) = 0.8 + 0.2 = 1, & x \ge 8

\end{cases}$

The graph of $F(x)$ is a step function with jumps at $x = 1, 2, 4, 8$.

3) Find the mode:

The mode is the value of $X$ with the highest probability. We have:

$P(X=1) = 0.3$

$P(X=2) = 0.4$

$P(X=4) = a = 0.1$

$P(X=8) = 2a = 0.2$

The highest probability is $0.4$, which corresponds to $X=2$.

Thus, the mode is

2.

4) Find $M(X)$, $D(X)$, and $\sigma(X)$:

$M(X) = \sum x_i P(X=x_i) = 1(0.3) + 2(0.4) + 4(0.1) + 8(0.2) = 0.3 + 0.8 + 0.4 + 1.6 = 3.1$

$M(X) = 3.1$

$D(X) = \sum (x_i - M(X))^2 P(X=x_i) = (1-3.1)^2(0.3) + (2-3.1)^2(0.4) + (4-3.1)^2(0.1) + (8-3.1)^2(0.2)$

$D(X) = (-2.1)^2(0.3) + (-1.1)^2(0.4) + (0.9)^2(0.1) + (4.9)^2(0.2) = 4.41(0.3) + 1.21(0.4) + 0.81(0.1) + 24.01(0.2)$

$D(X) = 1.323 + 0.484 + 0.081 + 4.802 = 6.69$

$D(X) = 6.69$

$\sigma(X) = \sqrt{D(X)} = \sqrt{6.69} \approx 2.5865$

3. Final Answer

1) $a = 0.1$

2) $F(x) = \begin{cases}

0, & x < 1 \\

0.3, & 1 \le x < 2 \\

0.7, & 2 \le x < 4 \\

0.8, & 4 \le x < 8 \\

1, & x \ge 8

\end{cases}$

3) Mode = 2

4) $M(X) = 3.1$, $D(X) = 6.69$, $\sigma(X) \approx 2.5865$