Question

5. Let X associated probabilities P(X = x)-/(2) be a discrete random variable. The following table shows its possible values r and the () 2/8 3/8 2/8 1/8 (a) Verify that f(x) is a probability mass function. (b) Calculate P(X < 1), P(X s 1), and P(X0.5 or Xx> 2) (c) Find the cumulative distribution function of X. (d) Compute the mean and the variance of X

please show you steps, and add some exppanation if possible. Thank you!

Answer 1

From the given information, x 1013 a) If the given f (x) is probability mass function, then follow the below rule: 2f(x)-1 2+3+2+1 Hence, the given f (x) satisfies the above rule, so, this is valid probability mass function.

b) Calculate the probability that P(X<1) P(x<l)-P(x so) 2+3 -0.625 Hence, the required probability is 106251.

Calculate the probability that P(X s1) 2+3+2 - 0.875 Hence, the required probability is 0.875

Calculate the probability that P(X<0.5 or X>2) P(X<0.5 or X> 2)- P(X<0.5 uX>2) -P(X < 0.5)+P(X > 2) +P(X < 0.5 X > 2) 2+3+1 =0.75 Hence, the required probability is0.75

c) Calculate the cumulative distribution function of X. f (x2/8 3/82/8 1/8 F(x) |-= 0.25-= 0.625-=0.875 |으= 1.00 Calculate the mean and variance of the X. Mean(X)-2xf(x) -2+0+2+3 0.375 Hence, the required mean is: [0.375.

Calculate the variance of the Variance(X)-E(X")-(E(X)) 864 (0.3- 2+0+2+9 (9 64 13 9 8 64 104-9 64 95 64 = 1.484 Hence, the required variance is: 1.484