end-to-end gamma = [camera gamma]*[display gamma]
Camera gamma has a 0.45 exponent but is simplified to 0.51(correction thanks to Andrewfee see his post below)
The highlights are effectively gamma 0.5, going from highlights to shadows the camera goes from 0.5 to 1.0 gamma, and at the same time the camera gain factor slowly increases from 1 to 4.5.
Reference monitors use during mastering use gamma 2.2 (PAL uses 2.35). But consumers are assumed to be using crt gamma 2.5 (PAL is mastered to look acceptable 2.2 to 2.5). Color might look best at 2.2 (PAL 2.35), but should look ok at 2.5, using gamma higher than 2.5 may result in problems with color on some material.
Perception of contrast is effected by the viewing condition surround effect so different gamma is usually recommended for different viewing conditions.
Gamma of 2.5 for dim surround conditions, dim is defined as being between 0% and 20% of the luminance of white in the image).
Gamma of 2 for bright surround conditions
Gamma of 3 for dark surround conditions
Video gamma is used in tv, dvd, blu-ray, etc... including for films. High gamma = more contrast in lighter part of image, less contrast in darker part. Low gamma = more contrast in darker part of image, less contrast in brighter part. The transition point between raised and lowered contrast is in the middle intensitys, not hidden in near black. Always above middle 18% grey (perceived as middle 50% brightness). The average image brightness will also be lower with higher gamma. So it comes down to personal choice visibility of shadow detail and a brighter image vs more image depth in bright scenes.
Also video has increased color saturation over film, because expected to be displayed on bright crt. Gamma effects color saturation and hue as well. As gamma is increased, dark colors at the bottom end of the gamma curve get dimmer, while brighter colors at the top end of the gamma curve get brighter, so a mix of colors with each at different points in the gamma curve, shift in ratio/proption to each other, the color mix gets more saturated since the lowest out of the rgb colors determines how much white - desaturation is present, and the hue shifts towards the color highest in the mix.
Film gamma is used only in commercial cinemas, not consumer versions like dvd or blu-ray, it uses a lazy s-curve. The s-curve transition point is darker, and gets darker the less gamma is increased, well below middle grey, 11% grey at most. The middle point gamma is typically 3.0. So for film higher gamma = more contrast, lower gamma = lower contrast. Sadly dvds, blu-rays are remastered for video altering their gamma and color so may not look acceptable displayed using film gamma that would otherwise be better for home cinema.