I have the standard deviation, mean and the maximum value for a distribution between t=0 and t=T. How do I approximate the best fitting curve? I have the time series data. The mean and maximum value, standard deviation at every minute (60s) is given. However, I want to intrapolate it as I want the data of each second. Is there any way to do this ?
 A: If you could provide an example dataset would be good. 
If I understand your question correctly, you have the mean value, standard deviation and maximum value for every minute and want to find an optimal way to interpolate to second data using this information?
Do you need to include the information on standard deviation you have? 
If not, you can use a simple interpolation, such as with Scipy.
Let's assume this is your original data set and the resulting averaged data:

then interpolating using only the mean values will result in this

looking at it on the original data

The code for this is below. 
If you provide more information on what kind of data you have and what you want to use it for, a more specific answer is possible.
    FIGSIZE=(10,4)
    import matplotlib.pylab as plt
    import seaborn as sns
    sns.set_style("whitegrid")
    plt.close('all')

    import numpy as np
    import scipy.signal

    #Create an example dataset
    T = 100 #Period
    t = np.arange(1200)
    y = np.sin(t/T) + 0.4*np.random.rand(len(t))

    plt.figure(figsize=FIGSIZE)
    plt.plot(t,y, lw=1, label='Original Data')
    plt.xlabel('Time')

    #Average data
    MINS = int(np.max(t)/60)
    tm=np.zeros(MINS); ym=np.zeros(MINS); ysd=np.zeros(MINS);         ymax=np.zeros(MINS)
    for ii in range(MINS):
        st = ii*60
        et = (ii+1)*60
        tempy = y[(t>st) & (t<et)]

        tm[ii] = ii+1
        ym[ii] = np.mean(tempy)
        ysd[ii] = np.std(tempy)
        ymax[ii] = np.max(tempy)

    plt.errorbar(tm*60 - 30, ym, yerr=ysd, marker='o', linestyle='-', capsize=5, 
                 elinewidth=2, label='Averaged Data')
    plt.legend(loc='best')
    plt.savefig('reduceddata.jpg', dpi=300)

    #From http://scipy-cookbook.readthedocs.io/items/Interpolation.html

    from scipy.signal import cspline1d, cspline1d_eval

    newt = np.arange(0,20,0.1)
    cj = cspline1d(ym)
    newy = cspline1d_eval(cj, newt, dx=1,x0=0)
    plt.figure(figsize=FIGSIZE)
    plt.errorbar(tm*60 - 30, ym, yerr=ysd, marker='o', linestyle='-', markersize=2,
                 lw=1, capsize=5, elinewidth=2, label='Averaged Data')

    plt.plot(newt*60+30, newy, 's-', markersize=3, label='Interpolation')
    plt.xlabel('Time')
    plt.show()
    plt.legend(loc='best')
    plt.savefig('interpolation.jpg', dpi=300)

    plt.figure(figsize=FIGSIZE)
    plt.plot(t, y, label='Original Data')
    plt.plot(newt*60+30, newy, 's-', markersize=4, label='Interpolation')
    plt.xlabel('Time')
    plt.legend(loc='best')
    plt.show()    
    plt.savefig('interpolation2.jpg', dpi=300)

