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- 3 The probability density function of a continuous random variable is given by cx, 0Q1 (a) A fair coin is tossed 5 times and the random variable X counts the number of headsthat come in the 5 tosses. Use the definition of expected value for discrete randomvariables to compute the expected value of X. (b) The continuous random variable T has f(t) = 2/t3t ≥ 10 t < 1as its probability densityfunction. Use the definition of expected value for continuous random variables tocompute the expected value of T.please solve c partDetermine E(X)(****) A continuous random variable X has probability density function (PDF) for some constants a and b. If Pr(X ≤ 0) is 2 times Pr(X > 0), what are a and b? a= b = f(x) = Jae for x < 0 (bxe-x² for x ≥ 0A random variable X has the following probability density function k, 0The probability density function of a random variable is kVx+ – 1 f(x) if 1< x< 2 = 0 otherwise Find the k. The marks obtained in mathematics by 1000 students are normally distributed with mean 78% and standard deviation 11%. Determine what was the highest mark obtaianed by the lowest 10% of the student? oninte The probability density function of a random variable is kx³ f(x) if 0sx<2 V25 – x² = 0 otherwise Find the k.Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropyof the random variable is defined asH(X) = −Σpilog (pi) Find the probability mass function for the above discrete random variable that maximizes the entropy.13.THEORY OF DISTRIBUTIONSSEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,