Problems 27: the number X is the number of car which passed through the specific stop sign in 1 minutes :
X 0 1 2 3 4 5

P(x) a 0.30 0.05 0.03 0.02 0.01

a) find out the value of a. (1pt)

b). the probability that there are at least 3 cars passing through the stop sign.
(1pt)

c). the expected value of X. (1pt)

d). the variance of X. (1pt)

e) the standard deviation of X. (1pt)

a) The value of a= 0.59

b)  The probability that there are at-least 3 cars passing through the stop sign

P(x>3)  = 0.03

c)

The Expected value of X  = 0.62

d)

 The variance of X is   σ² = 0.9556

e)

The standard deviation of X

                                    σ = 0.9775

Step-by-step explanation:

Given data

X       :      0         1          2            3        4          5

P(x)   :    a       0.30     0.05    0.03    0.02     0.01

a)

      ∑ = 1

 a + 0.30+ 0.05+ 0.03+ 0.02+0.01 = 1

 a  + 0.41 = 1

     a  = 1 - 0.41

     a  = 0.59

b)

 The probability that there are at-least 3 cars passing through the stop sign

P(x >3)  = P( x=4) + P( x=5)

           =   0.02 +0.01

          = 0.03

c)

X       :      0           1          2            3        4          5

P(x)   :      0.59      0.30     0.05    0.03    0.02     0.01

The Expected value of X

E(X) = ∑ x P(X= x)

      =  0 + 1 ×0.30 + 2×0.05 + 3×0.03 + 4×0.02 + 5×0.01

     =   0.30 + 0.1 + 0.09 +0.08 +0.05

    =  0.62

The Expected value of X

                          E(X)  = 0.62

d)

   The variance of the discrete distribution

       σ² =   ∑ x²p(x) -μ²

      σ² =   0 + 1² ×0.30 + 2² ×0.05 + 3² ×0.03 + 4² ×0.02 + 5²× 0.01 - (0.62)²

          =   1.34 - 0.3844

         = 0.9556

      σ² = √0.9556

e) The standard deviation of the  discrete distribution

     σ = 0.9775

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step-by-step explanation: