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TASK LIST NO. 3: Parameters of Random Variable Distribution

(Expected value, variance, moments, correlation)

Task 1

For a random variable \(X\) with the probability function given by the table:

\(x_i\) -2 2 4
\(p_i\) 0.5 0.3 0.2

Determine:

  1. The expected value \(E(X)\) (mean).
  2. The variance \(D^2(X)\) (using the formula \(D^2(X) = E(X^2) - (EX)^2\)).
  3. The standard deviation \(\sigma\).
  4. The median \(x_{0.5}\) (middle value).

Task 2

The monthly cost \(U\) of running a certain system depends on the number \(X\) of active users (employees) according to the formula:

\[ U = 15000X + 10000\sqrt{X} \]

The number of employees \(X\) is a random variable with the distribution:

\(x_i\) 2 3 4 5
\(p_i\) 0.10 0.25 0.40 0.25

Calculate the predicted average monthly cost, i.e., the expected value of the variable \(U\).

Hint: Calculate \(u_i\) for each \(x_i\), and then apply the formula for the expected value.

Task 3

A random variable \(X\) (e.g., measurement error) has a distribution with the density:

\[ f(x) = \begin{cases}6x(1-x) & \text{for } 0 < x < 1 \\0 & \text{for other } x\end{cases} \]

Calculate the average (expected) value and the variance of this variable. Then calculate the variance of the linearly dependent variable \(Y = 2X - 1\) (use the property of variance: \(D^2(aX+b) = a^2 D^2(X)\)).

Task 4

A random variable \(X\) has a distribution with the density:

\[ f(x) = \begin{cases}\frac{1}{2}x & \text{for } 0 \leqslant x \leqslant 2 \\0 & \text{otherwise}\end{cases} \]

Determine the mode (the value for which the density is greatest) and the median (the value that divides the area under the density graph into two equal halves).

Task 5

The height of people in a certain group is a random variable \(X\) with a mean \(EX = 170\) cm and a standard deviation \(\sigma_X = 5\) cm. The mass of these people is a variable \(Y\) with a mean \(EY = 65\) kg and a standard deviation \(\sigma_Y = 5\) kg.

Which feature (height or weight) is more "stable" (has a smaller relative dispersion)?

Hint: Calculate the coefficient of variation \(v = \frac{\sigma}{EX}\) for both variables.

Task 6

The probability of not exceeding the daily electricity consumption limit by a certain plant is \(p=0.8\). We observe this plant for \(n=5\) days. Let \(X\) denote the number of days in which the limit was not exceeded.

  1. What type of distribution is this? Provide the formula for the probability \(P(X=k)\).
  2. Calculate the expected value and variance of the variable \(X\), using the ready-made formulas for this distribution (\(EX=np\), \(D^2X=npq\)).

Task 7

The time (in minutes) between consecutive subscriber calls at a telephone exchange is a random variable with an exponential distribution with the parameter (expected value) \(\lambda = 2\).

  1. Calculate the average waiting time for a call (\(EX\)).
  2. Calculate the probability that the time between calls will be shorter than 3 minutes (\(P(X<3)\)).

Task 8

A machine produces weights. Mass measurement errors have a normal distribution with an expected value \(\mu=0\) g and a standard deviation \(\sigma=0.01\) g. Calculate the probability that the measurement error (in terms of modulus) does not exceed \(0.02\) g.

Hint: Use the cumulative distribution function of the standardized normal distribution \(\Phi(u)\). Note that \(P(|X|<a) = P(-a < X < a)\).

Task 9

Given is a two-dimensional random variable \((X, Y)\) with the distribution given in the table (representing, for example, test results at two different moments in time):

\(y_k \backslash x_i\) 8 9 10 11
1.2 0.10 0.04 0 0
1.3 0.05 0.11 0.20 0
1.4 0 0.10 0.15 0.10
1.5 0 0 0.05 0.10

Calculate the linear correlation coefficient \(\rho\) between variables \(X\) and \(Y\).

Hint: Calculate sequentially: means \(EX, EY\), variances \(D^2X, D^2Y\), and the mixed moment \(E(XY) = \sum x_i y_k p_{ik}\). Covariance is \(cov(X,Y) = E(XY) - EX \cdot EY\).

Task 10

Let \(X\) and \(Y\) be independent random variables with zero expected values (\(EX=0, EY=0\)). Show that the following equality holds:

\[E(X^3 Y) = E(X^3)E(Y)\]

Does the variable \(Z = X^3 Y\) have an expected value equal to 0? What does this mean in the context of random signals (noise)?