Question

Part 2. Random Variables 4. Two independent random variables Xand y are given with their distribution laws 0.3 0.7 0.8 0.2 Pi Find the distribution law and variance for the random variable V-3XY 5. There are 7 white balls and 3 red balls in a box. Balls are taken from the box without return at randomm until one white ball is taken. Construct the distribution law for the number of taken balls. 6. Let X be a continuous random variable whose probability density function is 0.5x'e-r, ja xe0 f(x) 0, Find 1) the cumulative distribution function; 2) P(X<2). 7. The random variable X is distributed exponentially with the parameter λ-07. Find the probability that the random variable X falls in the interval(-3; 2). Part 3. Statistics The following results are obtained after measuring of 12 elements (mm): 27.1; 27; 26.9; 27; 26.8; 26.9; 27.1; 27; 27.2; 27; 27.1; 26.9 8. Estimate the mean value of population, its variance and the standard deviation. 9. Find 95% confidence interval for the mean value

Answer 1

4) The distribution of the random variable $V=3XY$ is found as

P( V = 0) = P(X = 0) = 0.3 P(V = 12) = P(X = 4, Y 1) = P(X = 4)P(Y = 1) = 0.7(0.8) 0.56 P(V = 24) = P(X = 4, Y = 2) = P(X = 4)P(Y = 2) = 0.7(0.2) 0.14

Thus the distribution of $V=3XY$ is

V=3XY	0	12	24
P(V)	0.3	0.56	0.14	1.0

Now,

E(V) = 0(0.3) + 12(0.56) + 24(0.14) E(V) = 10.08 E(V2)-0(0.3)122(0.56) 242(0.14) E(V2) -161.28

The variance of the random variable is

Var(V)-E(V)- E(V) Var(V) 161.28-10.082 Var(V)59.6736

We are required to solve only one question. Please post the remaining questions as another post.