PS - nahodna velicina

nahodna_velicina.R

str_hodnota = function(v, p)
{
  sum(v * p);
}
rozptyl = function(v, p)
{
  str_hodnota(v ** 2, p) - str_hodnota(v, p) ** 2
}
var_koeficient = function(v, p)
{
  sqrt(rozptyl(v, p)) / abs(str_hodnota(v, p))
}

# 1.a
v = c(100, 200, 300, 500)
p = c(0.15, 0.3, 0.35, 0.2)
e = str_hodnota(v, p)
r = rozptyl(v, p)
sqrt(r)

# 1.b
vY = (100 - v) / 100
eY = str_hodnota(vY, p)
rY = rozptyl(vY, p)


# 2.a
p = c(0, 0, 0, 0, 0)
p[1] = choose(15, 4) / choose(20, 4)                    # vsechny panely OK
p[2] = (choose(15, 3) * choose(5, 1)) / choose(20, 4)   # jeden panel kaput
p[3] = (choose(15, 2) * choose(5, 2)) / choose(20, 4)   # dva panely kaput
p[4] = (choose(15, 1) * choose(5, 3)) / choose(20, 4)   # tri panely kaput
p[5] = choose(5, 4) / choose(20, 4)                     # ctyri panely kaput

p3 = sum(p[1:3])  # mene nez tri budou vadne


# 3.a
# c + 4c + 9c = 1
c = 1/14

# 3.b
# (-inf, 1> = 0
# (1, 2> = 1/14
# (2, 3> = 5/14
# (3, +inf) = 1

# 3.c
v = c(1,2,3)
p = c(1/14, 4/14, 9/14)
modus = 3
var = var_koeficient(v, p)

# 3.d
1/14          # P(X < 2)
p[1] + p[3]   # P(X == 1 || X == 3)


# 4.a
v = c(1, 2, 3, 4, 5, 6)
p = c(0, 0, 0, 0, 0, 0)
p[1] = 3/8        # modra koule bude prvni
p[2] = 5/8 * 3/7  # modra koule bude druha
p[3] = 5/8 * 4/7 * 3/6  # modra koule bude treti
p[4] = 5/8 * 4/7 * 3/6 * 3/5  # moudra koule bude ctvrta
p[5] = 5/8 * 4/7 * 3/6 * 2/5 * 3/4  # moudra koule bude pata
p[6] = 5/8 * 4/7 * 3/6 * 2/5 * 1/4  # modra koule bude sesta

# 4.b
modus = 1


# 5
ex = 1
ey = 3
dx = 4
dy = 9

ez = 4 * ex - 2 * ey + 12
dz = 16 * dx + 4 * dy
eq = -2 * ex + ey -7
dq = 4 * dx + dy


# 6.a
c = 3/8
# P(x) = 2 * x^2 - 2/3 * x^3

# 6.b
3/8 * 2t^2 - 2/3 * t^3

# 6.c
E = -1

# 6.d
0.6527

# 6.e
# P(X = 1) = 0
# P(X < 3) = 1
# P(X > 0.5) = 1 - 0.15625 (integral hustoty od 0 do 0.5)


# 7