Abstract. In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a 1 sinh x + a 2 sinh 2x + ··· + a n sinh nx, where a j (j = 1, 2,...,n) are independent normally distributed random variables with mean zero and variance one, is (1/π ) log n. This result is true for allIt is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a 1 cosh x + a 2 cosh 2x +···+a n cosh nx, with a j (j = 1, 2,...,n) as before, is also (1/π ) log n.
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